Hindawi Publishing CorporationISRN Probability and StatisticsVolume 2013 Article ID 543723 12 pageshttpdxdoiorg1011552013543723

Research ArticleTightness Criterion and Weak Convergence forthe Generalized Empirical Process in 119863[0 1]

Maciej Ziemba

Department of Mathematics Lublin University of Technology ul Nadbystrzycka 38d20-618 Lublin Poland

Correspondence should be addressed to Maciej Ziemba maciekziembagmailcom

Received 27 June 2013 Accepted 23 August 2013

Academic Editors M Campanino S Lototsky H J Paarsch and L Sacerdote

Copyright copy 2013 Maciej Ziemba This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We prove Shao and Yursquos tightness criterion for the generalized empirical process in the space 119863[0 1] with 1198691topology Covariance

inequalities are used in applying the criterion to particular types of the empirical processes We weaken the assumptions imposedon the covariance structure as well as the properties of the underlying sequence of rvrsquos under which presented processes convergeweakly

1 Introduction

Let 119883119899119899ge1

be a sequence of absolutely continuous identicallydistributed (id) random variables (rvrsquos) with an unknowndistribution function (df)119865 and probability density function(pdf) 119891 The empirical distribution function based on thefirst 119899 rvrsquos is defined by 119865

119899(119909) = 119899

minus1sum

119899

119895=1119868[119883

119895le 119909] It is

well known however that this estimate does not make useof the smoothness of 119865 that is the existence of the pdf 119891Therefore the kernel estimate

119865119899(119909) =

1

119899

119899

sum

119895=1

119870(

119909 minus 119883119895

ℎ119899

) (1)

has been proposed where the kernel function 119870 is a knowndf and ℎ

119899119899ge1

is a sequence of positive constants descendingat an appropriate rate Such estimator has been deeply studiedin the last two decades mainly by Cai and Roussas in [1ndash4]Li and Yang in [5] and others Asymptotic normality Berry-Essen bounds for smooth estimator 119865

119899(119909) are only examples

of their fruitful resultsRecently Li et al proposed in [6] the so-called recursive

kernel estimator of the df 119865 as follows

119865119899(119909) =

1

119899

119899

sum

119895=1

119870(

119909 minus 119883119895

ℎ119895

) (2)

The seemingly tiny modification they introduced to theformula of the typical kernel estimator has an importantadvantage Namely in the case of a large size of a sample119865119899(119909) can be easily updated with each new observation since

it is computable recursively by

119865119899(119909) =

119899 minus 1

119899119865119899minus1

(119909) +1

119899119870(

119909 minus 119883119899

ℎ119899

) (3)

where 1198650(119909) = 0 The authors discussed the asymptotic bias

and quadratic-mean convergence and established the point-wise asymptotic normality of 119865

119899(119909) under relevant assump-

tionsIn this paper however we will focus on the empirical

process built on an estimator 119865119899(119909) of the df 119865 rather than

119865119899(119909) itself Let us recall that the following process

120572119899(119909) = radic119899 [119865

119899(119909) minus 119864119865

119899(119909)] where 119909 isin R (4)

is called the empirical process built on an estimator 119865119899(119909)

Yu [7] studied the case when119865119899(119909) is a standard empirical

df and showed weak convergence of 120572119899(sdot) to the Gaussian

process assuming stationarity and association of the underly-ing rvrsquos Cai and Roussas [1] obtained a similar result in thecase when 119865

119899(119909) is the kernel estimator of the df 119865 built on

a stationary sequence of negatively associated rvrsquos

2 ISRN Probability and Statistics

In this paper we shall study the empirical process 120572119899(119909)

generated by the generalized kernel estimator of the df givenby the formula

119865119899(119909) =

1

119899

119899

sum

119895=1

119870(

119909 minus 119883119895

ℎ119899119895

) where (5)

A1 119883119895119895ge1

is a sequence of absolutely continuous id rvrsquostaking values in [0 1] and having twice differentiabledf 119865 with first and second derivative bounded

A2 119870 is a kernel function such that intR 119906119889119870(119906) = 0 andintR 119906

2119889119870(119906) lt infin with bounded derivative 119896

A3 ℎ119899119895119899ge1119895isin1119899

is a sequence of positive constantssubject to the following conditions lim

119895119899rarrinfinℎ119899119895

= 0lim

119895119899rarrinfin119899ℎ

4

119899119895= 0 (actually since 119895 le 119899 119895 rarr infin

under the limit suffices)

Explicitly we take a look onto the process

120572119899(119909) =

1

radic119899sdot

119899

sum

119895=1

[119870(

119909 minus 119883119895

ℎ119899119895

) minus 119864119870(

119909 minus 119883119895

ℎ119899119895

)] (6)

we shall call from now on the generalized empirical processLet us pay attention to the fact that in the case of

(i) ℎ119899119895

= ℎ119899for 119895 isin 1 119899 120572

119899(119909) is the empirical

process based on the kernel estimator of the df 119865(ii) ℎ

119895119895= ℎ

119895+1119895= ℎ

119895+2119895= sdot sdot sdot = ℎ

119895for 119895 isin N 120572

119899(119909)

is the empirical process based on the recursive kernelestimator of the df 119865

(iii) 119870(119905) = 119868[0infin]

(119905) 120572119899(119909) is the standard empirical

process (based on the empirical df)

It is well known that the crucial procedure in showingweak convergence for an empirical process is to verify tight-ness In [8] Shao and Yu gave the following criterion

exist1198621gt0exist

119901gt2exist

1199011gt1exist

0le1199031le1exist

1199012gt1minus1199031

forall119909119910isin[01]

1198641003816100381610038161003816120572119899

(119909) minus 120572119899(119910)

1003816100381610038161003816

119901

le 1198621(1003816100381610038161003816119909 minus 119910

1003816100381610038161003816

1199011

+ 119899minus119901221003816100381610038161003816119909 minus 119910

1003816100381610038161003816

1199031

)

(7)

under which the standard empirical process based on sta-tionary sequence of uniform [0 1] rvrsquos is tight It is statedthere that the proof of that fact is an easy standard procedureparallel to the one presented in [9] It is the main aim of thispaper to carry it in details but for the generalized empiricalprocess defined by (6) and without assuming stationarityNevertheless wewill always return to stationarity assumptionwhile establishing weak convergence

In order to obtain tightness one has to assume appro-priate covariance structure of the underlying rvrsquos that isthe covariance of a pair of rvrsquos 119883

119894and 119883

119895has to decline

at the right rate while 119894 and 119895 are growing apart In thispaper we lower the demanded rate of covariance decay usingthe covariance inequalities for associated (cf [10 11]) andmultivariate totally positive of order 2 (MTP

2) (cf [12]) rvrsquos

obtained in [13]

The paper is organized as follows In Section 2 we presentthe proof of the Shao and Yursquos tightness criterion formulatedfor our generalized empirical process Sections 3 and 4 aredevoted to application of the criterion to showing tightnessand thus weak convergence of the specific types of empiricalprocesses Section 5 concerns weak convergence of the recur-sive kernel-type process for iid rvrsquos

2 Tightness Criterion

We start with the key point of the paper

Theorem 1 Let 120572119899(119909)

119899ge1isin 119863[0 1] be the generalized empir-

ical process defined as in (6) One assume that A1 A2 A3 holdIf there exist constants 119862 gt 0 119901 gt 2 119901

1gt 1 0 le 119901

3le 1

1199012gt 1 minus 119901

3 such that for any 119909 119910 isin [0 1] and 119899 isin N the

following inequality holds

1198641003816100381610038161003816120572119899

(119909) minus 120572119899(119910)

1003816100381610038161003816

119901

le 119862 sdot (1003816100381610038161003816119909 minus 119910

1003816100381610038161003816

1199011

+ 119899minus11990122sdot1003816100381610038161003816119909 minus 119910

1003816100381610038161003816

1199013

)

(8)

then the process 120572119899(119909)

119899ge1is tight in119863[0 1] with 119869

1topology

Proof The proof boils down to showing that under theassumptions made inTheorem 1 conditions ofTheorem 132in [9] hold Let us recall that in light of the above mentionedtheorem a process 120572

119899(119909)

119899ge1is tight in119863[0 1] if

lim119886rarrinfin

lim sup119899rarrinfin

119875 (1003817100381710038171003817120572119899

1003817100381710038171003817 ge 119886) = 0 (9)

lim120575rarr0

lim sup119899rarrinfin

119875 (1199081015840

1(120572

119899 120575) ge 120598) = 0 forall120598 gt 0 (10)

where sdot is the supremum norm that is1003817100381710038171003817120572119899

1003817100381710038171003817 = sup0le119909le1

1003816100381610038161003816120572119899(119909)

1003816100381610038161003816 (11)

and 1199081015840

1(120572

119899 120575) is the modulus of continuity of the function

120572119899isin 119863[0 1] that is

1199081015840

1(120572

119899 120575) = inf

119905119894

max1le119894le]

sup119909119910isin[119905119894minus1119905119894)

1003816100381610038161003816120572119899(119909) minus 120572

119899(119910)

1003816100381610038161003816 (12)

The infimum runs over all finite ldquo120575-sparserdquo decompositions[119905

119894minus1 119905

119894) 119894 isin 1 ] of the interval [0 1] In other words it

runs over all choices of increasingly ordered points 1199051198941le119894le]

such that min1le119894le](119905119894 minus 119905

119894minus1) gt 120575 where 119905

0= 0 119905] = 1

Let us first show that condition (9) is satisfied Accordingto the corollary following Theorem 132 in [9] it suffices toshow

lim119886rarrinfin

lim sup119899rarrinfin

119875 (1205721198991003816100381610038161003816120572119899

(119909)1003816100381610038161003816 ge 119886) = 0 forall119909 isin [0 1] (13)

that is

forall120578ge0

exist119886gt0

forall119899isinN119875 (120572

1198991003816100381610038161003816120572119899

(119909)1003816100381610038161003816 ge 119886) le 120578 forall119909 isin [0 1] (14)

Let us fix 119909 isin [0 1] and 120578 gt 0 We need to find 119886 = 119886(120578) gt 0such that

119875 (radic11989910038161003816100381610038161003816119865119899(119909) minus 119864119865

119899(119909)

10038161003816100381610038161003816ge 119886) le 120578 forall119899 isin N (15)

ISRN Probability and Statistics 3

where

119865119899(119909) =

1

119899

119899

sum

119895=1

119870(

119909 minus 119883119895

ℎ119899119895

) (16)

is the generalized kernel estimator of the df119865 ApplyingChe-byshevrsquos inequality we shall find 119886 satisfying

119886 geradic119899

120578sdot 119864

10038161003816100381610038161003816119865119899(119909) minus 119864119865

119899(119909)

10038161003816100381610038161003816forall119899 isin N (17)

Such 119886 exists if 119864|120572119899(119909)| lt infin for all 119899 ge 1 Implementing

119910 = 0 and 1199013= 0 in the assumption (8) we get for 119901 gt 2

1199011 119901

2gt 1

1198641003816100381610038161003816120572119899

(119909)1003816100381610038161003816

119901

le 119862 sdot(|119909|1199011+119899

minus11990122|119909|

1199013) le 119862 sdot(1+

1

11989911990122) le 2119862

(18)

Now applying Holderrsquos inequality we arrive at

1198641003816100381610038161003816120572119899

(119909)1003816100381610038161003816 le

4radic119864

1003816100381610038161003816120572119899(119909)

1003816100381610038161003816

4

(19)

which in light of the inequality (18) for 119901 = 4 is boundedfrom above by 4radic2119862 lt infin Therefore condition (9) holds

We shall now proceed to checking condition (10) Let usrecall that the modulus of continuity of the function 119909 isin

119862[0 1] 120575 gt 0 is given by the formula

1199081(119909 120575) = sup

0le119910le1minus120575

sup119910le119909le119910+120575

1003816100381610038161003816120572119899(119909) minus 120572

119899(119910)

1003816100381610038161003816 (20)

As it is shown in [9] (page 123) there is a relation between119908

1(sdot sdot) and 1199081015840

1(sdot sdot) in the spaces 119862[0 1] and119863[0 1] relatively

Namely for 119909 isin 119863[0 1] and 120575 lt 12

1199081(119909 2120575) ge 119908

1015840

1(119909 120575) (21)

Thus

119875 (1199081015840

1(120572

119899 120575) ge 120598) le 119875 (119908

1(120572

119899 2120575) ge 120598) (22)

Therefore condition (10) holds if we show

forall120598gt0

forall120578gt0

exist0le120575lt12

exist1198990

forall119899ge1198990

119875 (1199081(120572

119899 2120575) ge 120598) le 120578 (23)

With a view to obtaining the desired inequality we shall pro-ceed patiently in five steps

Step 1 We need a moment inequality for the rv 120572119899(119909)minus120572

119899(119910)

involving the distance between the points 119909 and 119910 Let usthen assume that for the constants 119862 119901 119901

1 119901

2 119901

3given in

Theorem 1 inequality (8) holds Fix 120598 gt 0 120578 gt 0 and definethe quantity 119903

119899= 120598radic119899 119899 isin N Next we fix 119909 119910 isin [0 1] and

take 119899 large enough so that 119903119899le |119909minus119910|Then 119899minus12

le |119909minus119910|120598minus1

and we have

1198641003816100381610038161003816120572119899

(119909) minus 120572119899(119910)

1003816100381610038161003816

119901

le 119862 (1 + 120598minus1199012)1003816100381610038161003816119909 minus 119910

1003816100381610038161003816

min11990111199012+1199013

(24)

Step 2 Let us now fix 119899 isin N 120575 isin (0 1) and consider the fol-lowing rvrsquos

120594119894= 120572

119899(119910 + 119894119903

119899) minus 120572

119899(119910 + (119894 minus 1) 119903

119899) (25)

for 119894 isin 1 2 119898119899 where119898

119899= 119898(119899 120575) is such that 119903

119899119898

119899le 120575

It is easy to see that for 119878119894= sum

119894

119896=1120594

119896

max1le119894le119898

119899

1003816100381610038161003816119878119894

1003816100381610038161003816 = max1le119894le119898

119899

1003816100381610038161003816120572119899(119910 + 119894119903

119899) minus 120572

119899(119910)

1003816100381610038161003816 (26)

Let us notice that for rvrsquos 120594119894119894ge1

the conditions of Theo-rem 102 in [9] are satisfied with 4120573 = 119901 119906

119897= 119903

119899forall

119894lt119897le119895and

2120572 = min1199011 119901

2+ 119901

3 Therefore we are equipped with the

following maximal inequality

119875( max1le119894le119898

119899

1003816100381610038161003816120572119899(119910 + 119894119903

119899) minus 120572

119899(119910)

1003816100381610038161003816 ge 120582)

le

119862119901min119901

11199012+1199013

120582119901(119898

119899119903119899)min119901

11199012+1199013

(27)

for all 120582 gt 0

Step 3 Let 119872 = sup119909isin[01]

119891(119909) where 119891 is the pdf of rvrsquos119883

119895119895ge1

For fixed 120598 120578 gt 0 let us take 120575 gt 0 such that

119862119901min119901

11199012+1199013

(119872120598)119901

sdot (2120575)min119901

11199012+1199013lt 120578 (28)

and define119898119899= lfloor120575119903

119899rfloor For sufficiently large 119899 isin N we have

119898119899ge 1 and then we get

119903119899119898

119899le 120575 lt (119898

119899+ 1) 119903

119899le 2119898

119899119903119899le 120575 (29)

Step 4 Our goal is to obtain an inequality which enables usto bound the supremum of the increment of the function 120572

119899

via the maximum of the increments of that function on somesubintervals To bemore precise wewill find the upper boundfor

sup119910le119909le119910+119898

119899119902

1003816100381610038161003816120572119899(119909) minus 120572

119899(119910)

1003816100381610038161003816 (30)

in terms of

max1le119894le119898

119899

1003816100381610038161003816120572119899(119910 + 119894119902) minus 120572

119899(119910)

1003816100381610038161003816 (31)

where 119909 119910 isin [0 1] 119910 le 119909 le 119910 + 119902 119902 ge 120598119899 and119898119899is defined

as in Step 3Let us recall that 120572

119899(119909) = radic119899(119865

119899(119909) minus 119864119865

119899(119909)) where

119865119899(119909) = (1119899)sum

119899

119895=1119870((119909 minus 119883

119895)ℎ

119899119895) From the triangle ine-

quality we can see that

1003816100381610038161003816120572119899(119909) minus 120572

119899(119910)

1003816100381610038161003816

leradic119899

119899

10038161003816100381610038161003816100381610038161003816100381610038161003816

119899

sum

119895=1

[119870(

119909 minus 119883119895

ℎ119899119895

) minus 119870(

119910 minus 119883119895

ℎ119899119895

)]

10038161003816100381610038161003816100381610038161003816100381610038161003816

+ radic119899 (10038161003816100381610038161003816119864119865

119899(119909) minus 119865 (119909)

10038161003816100381610038161003816+10038161003816100381610038161003816119864119865

119899(119910) minus 119865 (119910)

10038161003816100381610038161003816

+1003816100381610038161003816119865 (119909) minus 119865 (119910)

1003816100381610038161003816 )

(32)

4 ISRN Probability and Statistics

Since 119870 is a nondecreasing function and applying triangleinequality again we get

1003816100381610038161003816120572119899(119909) minus 120572

119899(119910)

1003816100381610038161003816

le1003816100381610038161003816120572119899

(119910 + 119902) minus 120572119899(119910)

1003816100381610038161003816 +radic119899

10038161003816100381610038161003816119864119865

119899(119910 + 119902) minus 119864119865

119899(119910)

10038161003816100381610038161003816

+ radic119899 (10038161003816100381610038161003816119864119865

119899(119909) minus 119865 (119909)

10038161003816100381610038161003816+10038161003816100381610038161003816119864119865

119899(119910) minus 119865 (119910)

10038161003816100381610038161003816

+1003816100381610038161003816119865 (119909) minus 119865 (119910)

1003816100381610038161003816 )

(33)

Cai and Roussas in [1] showed that under assumptions madeon the df 119865 and the kernel function119870 we have

10038161003816100381610038161003816119864119865

119899(119909) minus 119865 (119909)

10038161003816100381610038161003816= 119874 (ℎ

2

119899) (34)

Similarly by Taylor expansion it is easy to see that

1003816100381610038161003816119865 (119909) minus 119865 (119910)1003816100381610038161003816 le

10038171003817100381710038171198911003817100381710038171003817 sdot

1003816100381610038161003816119909 minus 1199101003816100381610038161003816 (35)

Thus

1003816100381610038161003816120572119899(119909) minus 120572

119899(119910)

1003816100381610038161003816 le1003816100381610038161003816120572119899

(119910 + 119902) minus 120572119899(119910)

1003816100381610038161003816

+ 119862radic119899ℎ4

119899+119872radic119899119902

(36)

where 119872 = sup119909isin[01]

119891(119909) and 119862 = 1198621198911015840119870

is a positive con-stant dependent on the functions 119865 and119870

Let us recall that 119910 le 119909 le 119910+ 119902119898119899= lfloor120575119903

119899rfloor and 119902 ge 120598119899

Taking 119902 = (120598radic119899)(= 119903119899) we notice that 119898

119899119902 = lfloor120575119903

119899rfloor119903

119899le 120575

thus when 119899 rarr infin the interval [119910 119910 + 119898119899119902] coincides with

[119910 119910 + 120575] Approaching the aim of Step 4 let us observe that

sup119910le119909le119910+119898

119899119902

1003816100381610038161003816120572119899(119909) minus 120572

119899(119910)

1003816100381610038161003816 = max1le119894le119898

119899

1003816100381610038161003816120572119899(119910 + 119894119902) minus 120572

119899(119910)

1003816100381610038161003816

+1003816100381610038161003816120572119899

(1199090) minus 120572

119899(119910 + 119894max119902)

1003816100381610038161003816

(37)

where 1199090isin [119910+119894

0119902 119910+(119894

0+1)119902] is a point at which the above

supremum is attained and

119894max =

1198940+ 1 if 1003816100381610038161003816120572119899

(119910 + [1198940+ 1] 119902) minus 120572

119899(119910)

1003816100381610038161003816

ge1003816100381610038161003816120572119899

(119910 + 1198940119902) minus 120572

119899(119910)

1003816100381610038161003816

1198940

otherwise(38)

Without the loss of generality we may assume that 119894max = 1198940+

1 which implies

sup119910le119909le119910+119898

119899119902

1003816100381610038161003816120572119899(119909) minus 120572

119899(119910)

1003816100381610038161003816 = max1le119894le119898

119899

1003816100381610038161003816120572119899(119910 + 119894119902) minus 120572

119899(119910)

1003816100381610038161003816

+1003816100381610038161003816120572119899

(1199090) minus 120572

119899(119910 + 119894

0119902)1003816100381610038161003816

(39)

Now applying inequality (36) the definition of 119894max and thetriangle inequality we have

sup119910le119909le119910+119898

119899119902

1003816100381610038161003816120572119899(119909) minus 120572

119899(119910)

1003816100381610038161003816

le max1le119894le119898

119899

1003816100381610038161003816120572119899(119910 + 119894119902) minus 120572

119899(119910)

1003816100381610038161003816

+1003816100381610038161003816120572119899

(119910 + [1198940+ 1] 119902) minus 120572

119899(119910 + 119894

0119902)1003816100381610038161003816

+ 119872radic119899119902 + 119862radic119899ℎ4

119899

le max1le119894le119898

119899

1003816100381610038161003816120572119899(119910 + 119894119902) minus 120572

119899(119910)

1003816100381610038161003816

+1003816100381610038161003816120572119899

(119910 + [1198940+ 1] 119902) minus 120572

119899(119910)

1003816100381610038161003816

+1003816100381610038161003816120572119899

(119910 + 1198940119902) minus 120572

119899(119910)

1003816100381610038161003816 + 119872radic119899119902 + 119862radic119899ℎ4

119899

le 3 max1le119894le119898

119899

1003816100381610038161003816120572119899(119910 + 119894119902) minus 120572

119899(119910)

1003816100381610038161003816 + 119872radic119899119902 + 119862radic119899ℎ4

119899

(40)

If we plug in 119902 = 119903119899we arrive at

sup119910le119909le119910+119898

119899119903119899

1003816100381610038161003816120572119899(119909) minus 120572

119899(119910)

1003816100381610038161003816 le 3 max1le119894le119898

119899

1003816100381610038161003816120572119899(119910 + 119894119903

119899) minus 120572

119899(119910)

1003816100381610038161003816

+ 119872radic119899119903119899+ 119862radic119899ℎ4

119899

(41)

Step 5 Finally we are in a position to obtain inequality (23)that is

forall120598gt0forall

120578gt0exist

0le120575lt12exist

1198990

forall119899ge1198990

119875( sup0le119910le1minus2120575

sup119910le119909le119910+2120575

1003816100381610038161003816120572119899(119909)minus120572

119899(119910)

1003816100381610038161003816

ge 120598) le 120578

(42)

We now successivelymake use the inequalities (29) (41) (27)(29) and (28) to get

119875( sup119910le119909le119910+2120575

1003816100381610038161003816120572119899(119909) minus 120572

119899(119910)

1003816100381610038161003816 ge 4119872120598 + 119862radic119899ℎ4

119899)

le 119875( sup119910le119909le119910+2(119898

119899+1)119903119899

1003816100381610038161003816120572119899(119909) minus 120572

119899(119910)

1003816100381610038161003816 ge 4119872120598 + 119862radic119899ℎ4

119899)

le 119875(3 max1le119894le2(119898119899+1)

1003816100381610038161003816120572119899(119910 + 119894119903

119899) minus 120572

119899(119910)

1003816100381610038161003816

+119872radic119899119903119899+ 119862radic119899ℎ4

119899ge 4119872120598 + 119862radic119899ℎ4

119899)

ISRN Probability and Statistics 5

= 119875( max1le119894le2(119898

119899+1)

1003816100381610038161003816120572119899(119910 + 119894119903

119899) minus 120572

119899(119910)

1003816100381610038161003816 ge 119872120598)

le

119862119901min119901

11199012+1199013

(119872120598)119901

(2120575)min119901

11199012+1199013le 120578

(43)

for any fixed 119910 isin [0 1] Since the upper bound does notdepend on 119910 and the probability measure as well as supre-mum function are continuous we obtain

119875( sup0le119910le1minus2120575

sup119910le119909le119910+2120575

1003816100381610038161003816120572119899(119909) minus 120572

119899(119910)

1003816100381610038161003816 ge 4119872120598 + 119862radic119899ℎ4

119899) le 120578

(44)

where 4119872120598 + 119862radic119899ℎ4

119899is arbitrarily small

Since condition (10) is checked the proof is completed

3 Tightness of the Standard Empirical Process

In this section we deal with the standard empirical processbuilt on an associated sequence of uniformly [0 1]distributedrvrsquos 119880

119895119895ge1

that is

120572119899(119909) =

1

radic119899

119899

sum

119895=1

(119868 [119880119895le 119909] minus 119909) (45)

where 119909 isin [0 1] We shall relax the restrictions imposedon the process by Yu in [7] to obtain tightness Preciselywe do not need stationarity any more due to the techniquedrawn from [14] and we lower the assumed rate at which thecovariance tends to zero

While proving tightness of the empirical process we willuse the criterion proved in the first section as well as some ofour covariance inequalities

We shall start with the fact known under the name ofmultinomial theorem We recall it in the following lemma

Lemma 2 For natural numbers119898 119899 and 1199091 119909

2 119909

119898isin R

(1199091+ sdot sdot sdot + 119909

119899)119898

= sum

1198961+sdotsdotsdot+119896

119899=119898

(119898

1198961 119896

119899

) prod

1le119894le119899

119909119896119894

119894 (46)

where (119898

1198961 119896

119899) = 119898(119896

1 sdot sdot sdot 119896

119899) and 119896

1 119896

119899isin

0 1 119898

In particular

(1199091+ sdot sdot sdot + 119909

119899)4

le 4 sdot sum

1198961+sdotsdotsdot+119896

119899=4

1199091198961

1sdot sdot sdot 119909

119896119899

119899 (47)

which implies for 119878119899= sum

119899

119894=1119883

119894 that

1198641198784

119899le 4 sdot sum

1198961+sdotsdotsdot+119896

119899=4

119864 (1198831198961

1sdot sdot sdot 119883

119896119899

119899)

le 4 sdot

119899

sum

119905=1

sum

0le119894119895119896le119899minus1

10038161003816100381610038161003816119864 (119883

119905119883

119905+119894119883

119905+119894+119895119883

119905+119894+119895+119896)10038161003816100381610038161003816

(48)

Let us now introduce the following notation due to Doukhanand Louhichi (see [14]) for a sequence of centered rvrsquos119883

119905119899

119899ge1

119862119903119902

= sup100381610038161003816100381610038161003816Cov (119883

1199051

sdot sdot sdot 119883119905119898

119883119905119898+119903sdot sdot sdot 119883

119905119902

)100381610038161003816100381610038161003816 (49)

where the supremum runs over all divisions of the groupcomposed of 119902 rvrsquos into two subgroups such that the distancebetween the highest index of the rvrsquos in the first group and thelowest index of the rvrsquos from the second group is equal to 119903119903 isin 1 2 For 119903 = 0 we shall define 119862

0119902= 1 Let us then

put

1198831199051

= 119868 [1198801199051

le 119909] minus 119909 1198831199052

= 119868 [1198801199052

le 119909] minus 119909 (50)

We will now estimate the summands in (48)If max119894 119895 119896 = 119894 then

10038161003816100381610038161003816119864 119883

119905sdot (119883

119905+119894119883

119905+119894+119895119883

119905+119894+119895+119896)10038161003816100381610038161003816

=100381610038161003816100381610038161003816Cov (119883

119905 119883

119905+119894119883

119905+119894+119895119883

119905+119894+119895+119896)100381610038161003816100381610038161003816le 119862

1198944

(51)

Similarly for max119894 119895 119896 = 119896 we have

10038161003816100381610038161003816119864 (119883

119905119883

119905+119894119883

119905+119894+119895) sdot 119883

119905+119894+119895+11989610038161003816100381610038161003816

=10038161003816100381610038161003816Cov (119883

119905119883

119905+119894119883

119905+119894+119895 119883

119905+119894+119895+119896)10038161003816100381610038161003816le 119862

1198964

(52)

When max119894 119895 119896 = 119895 then

10038161003816100381610038161003816119864 (119883

119905119883

119905+119894) sdot (119883

119905+119894+119895119883

119905+119894+119895+119896)10038161003816100381610038161003816

=10038161003816100381610038161003816Cov (119883

119905119883

119905+119894 119883

119905+119894+119895119883

119905+119894+119895+119896)

+119864 (119883119905119883

119905+119894) 119864 (119883

119905+119894+119895119883

119905+119894+119895+119896)10038161003816100381610038161003816

le 1198621198954

+ 1198621198942119862

1198962

(53)

We keep on estimating the fourth moment of 119878119899by intro-

ducing 119905 isin 0 119899mdashthe index for which |119864(119883119905119883

119905+119894119883

119905+119894+119895

119883119905+119894+119895+119896

)| attains its maximum

1198641198784

119899le 4119899 sum

0le119894119895119896le119899minus1

10038161003816100381610038161003816119864 (119883

119905119883

119905+119894119883

119905+119894+119895119883

119905+119894+119895+119896)10038161003816100381610038161003816

=4119899( sum

0le119894119896le119895

[1198621198942sdot 119862

1198962+ 119862

1198954]+ sum

0le119894119895le119896

1198621198964+ sum

0le119896119895le119894

1198621198944)

= 4119899( sum

0le119894119896le119895

1198621198942sdot 119862

1198962+ 3 sum

0le119896119895le119894

1198621198944)

(54)

6 ISRN Probability and Statistics

The terms obtained in (54) may further be bounded fromabove in the following way

sum

0le119894119896le119895

1198621198942sdot 119862

1198962=

119899minus1

sum

119895=0

119895

sum

119894119896=0

1198621198942119862

1198962

le 119899

119899minus1

sum

119894119896=0

1198621198942119862

1198962= 119899(

119899minus1

sum

119894=0

1198621198942)

2

sum

0le119896119895le119894

1198621198944

=

119899minus1

sum

119894=0

119894

sum

119895119896=0

1198621198944

le

119899minus1

sum

119894=0

(119894 + 1) (119894 + 1) 1198621198944

=

119899minus1

sum

119894=0

(119894 + 1)2119862

1198944

(55)

We thus get the inequality

1198641198784

119899le 4([119899

119899minus1

sum

119894=0

1198621198942]

2

+ 3119899

119899minus1

sum

119894=0

(119894 + 1)2119862

1198944) (56)

We shall now focus on estimating 1198621199032

and 1198621199034 Since 119880

1199051

and 1198801199052

are associated uniformly [0 1] distributed rvrsquos 1198831199051

and 1198831199052

mdashas monotone functions of these rvrsquosmdashare associat-ed as well In order to bound

1198621199032

= sup0le1199051lt1199052le119899 1199052minus1199051=119903

Cov (1198831199051

1198831199052

) (57)

let us notice that from Schwarz inequality

Cov (1198831199051

1198831199052

) = 119864 (119868 [1198801199051

le 119909] 119868 [1198801199052

le 119909]) minus 1199092

le radic1199092 minus 1199092le 119909

(58)

On the other hand invoking inequalities from [13 15]

Cov (1198831199051

1198831199052

)

le sup119909isin[01]

[119875 (1198801199051

le 1199091198801199052

le 119909) minus 119875 (1198801199051

le 119909)119875 (1198801199052

le 119909)]

le 119862 sdot Cov13

(1198801199051

1198801199052

) for associated rvrsquos4 sdot Cov (119880

1199051

1198801199052

) for MTP2rvrsquos

(59)

As a consequence

1198621199032

le min 119909 119862 sdot Cov13

(1198801199051

1198801199052

) for associated rvrsquosmin 119909 4 sdot Cov (119880

1199051

1198801199052

) for MTP2rvrsquos

(60)

where 1199052minus 119905

1= 119903 Still we need the upper bound for 119862

1199034 It

turns out that119862

1199034

= sup1003816100381610038161003816100381610038161003816100381610038161003816

Cov2

prod

119894=1

(119868 [119880119905119894

le 119909] minus 119909)

4

prod

119894=3

(119868 [119880119905119894

le 119909] minus 119909)

1003816100381610038161003816100381610038161003816100381610038161003816

(61)

In other words among all divisions of the group 119883119905119894

119894 isin

1 2 3 4 into two subgroups 1198621199034

attains the biggest valuein case we take two subgroups consisted of two rvrsquos Thesupremum runs over the set 119905

1 119905

2 119905

3 119905

4isin N 0 le 119905

1lt 119905

2lt

1199053lt 119905

4le 119899 and 119905

3minus 119905

2= 119903 Elementary calculation leads to the

following formula

1198621199034

= sup 10038161003816100381610038161003816119875 (119880119905119894

le 119909 119894 isin 1 2 3 4)

minus 119875 (119880119905119894

le 119909 119894 isin 1 2) 119875 (119880119905119894

le 119909 119894 isin 3 4)

minus 119909 [119875 (119880119905119894

le 119909 119894 isin 1 2 3)

minus 119875 (119880119905119894

le 119909 119894 isin 1 2) 119875 (1198801199053

le 119909)

+ 119875 (119880119905119894

le 119909 119894 isin 1 3 4)

minus 119875 (119880119905119894

le 119909 119894 isin 3 4) 119875 (1198801199051

le 119909)

+ 119875 (119880119905119894

le 119909 119894 isin 1 2 4)

minus 119875 (119880119905119894

le 119909 119894 isin 1 2) 119875 (1198801199054

le 119909)

+ 119875 (119880119905119894

le 119909 119894 isin 2 3 4)

minus119875 (119880119905119894

le 119909 119894 isin 3 4) 119875 (1198801199052

le 119909)]

+ 1199092[119875 (119880

1199051

le 1199091198801199053

le 119909)

minus 119875 (1198801199051

le 119909)119875 (1198801199053

le 119909)

+ 119875 (1198801199051

le 1199091198801199054

le 119909)

minus 119875 (1198801199051

le 119909)119875 (1198801199054

le 119909)

+ 119875 (1198801199052

le 1199091198801199053

le 119909)

minus 119875 (1198801199052

le 119909)119875 (1198801199053

le 119909)

+ 119875 (1198801199052

le 1199091198801199054

le 119909)

minus119875 (1198801199052

le 119909)119875 (1198801199052

le 119909)]10038161003816100381610038161003816

le10038161003816100381610038161198771

1003816100381610038161003816 +10038161003816100381610038161198772

1003816100381610038161003816 +10038161003816100381610038161198773

1003816100381610038161003816

(62)

where for the sake of simplicity 1198771 119877

2 and 119877

3are respec-

tively the free coefficient the expression with 119909 and theexpression with 119909

2 Using the Lebowitz inequality (see [15]for instance) and inequalities obtained in [13 15] we arrive at10038161003816100381610038161198771

1003816100381610038161003816 =10038161003816100381610038161003816119875 (119880

119905119894

le 119909 119894 isin 1 2 3 4)

minus119875 (119880119905119894

le 119909 119894 isin 1 2) 119875 (119880119905119894

le 119909 119894 isin 3 4)10038161003816100381610038161003816

le 11986711988011990511198801199053

+ 11986711988011990511198801199054

+ 11986711988011990521198801199053

+ 11986711988011990521198801199054

ISRN Probability and Statistics 7

le

119862(

2

sum

119894=1

4

sum

119895=3

Cov13(119880

119905119894

119880119905119895

)) for associated rvrsquos

4(

2

sum

119894=1

4

sum

119895=3

Cov (119880119905119894

119880119905119895

)) for MTP2rvrsquos

(63)

Analogously we get

10038161003816100381610038161198772

1003816100381610038161003816

le

2119909 sdot 119862(

2

sum

119894=1

4

sum

119895=3

Cov13(119880

119905119894

119880119905119895

)) for associated rvrsquos

2119909 sdot 4(

2

sum

119894=1

4

sum

119895=3

Cov (119880119905119894

119880119905119895

)) for MTP2rvrsquos

10038161003816100381610038161198773

1003816100381610038161003816

le

1199092sdot 119862(

2

sum

119894=1

4

sum

119895=3

Cov13(119880

119905119894

119880119905119895

) ) for associated rvrsquos

1199092sdot 4(

2

sum

119894=1

4

sum

119895=3

Cov (119880119905119894

119880119905119895

)) for MTP2rvrsquos

(64)

Eventually we have

1198621199034

le

119891 (119909) sdot119862(

2

sum

119894=1

4

sum

119895=3

Cov13(119880

119905119894

119880119905119895

)) for associated rvrsquos

119891 (119909) sdot4(

2

sum

119894=1

4

sum

119895=3

Cov (119880119905119894

119880119905119895

)) for MTP2rvrsquos

(65)

where 1199053minus 119905

2= 119903 and 119891(119909) = (1 + 2119909 + 119909

2) for 119909 isin [0 1]

Let us now introduce the following notation

120579119903=

sup119905isinN

Cov (119880119905 119880

119905+119903) for 119903 isin 1 2

1 for 119903 = 0

(66)

and assume it decays powerly at rate 119886 in the following way

120579119903= 119874(

1

(119903 + 1)119886) for 119886 gt 0 119903 isin 0 1 (67)

Let us get back to inequality (56) we can now carry on Atfirst for associated rvrsquos

1198641198784

119899le 4[119899

119899minus1

sum

119903=0

min 119909 119862 sdot Cov13(119880

1199051

1198801199052

)]

2

+ 4 sdot 3119899

119899minus1

sum

119903=0

(119903 + 1)2119891 (119909) sdot 119862(

2

sum

119894=1

4

sum

119895=3

Cov13(119880

119905119894

119880119905119895

))

le 119863 sdot ([119899

119899minus1

sum

119903=0

min 119909 (119903 + 1)minus1198863

]

2

+119899

119899minus1

sum

119903=0

(119903 + 1)2(119903 + 1)

minus1198863)

= 119863 sdot ([119899

119899

sum

119903=1

min 119909 119903minus1198863]

2

+ 119899

119899

sum

119903=1

1199032119903minus1198863

)

le 119863 sdot ([

[

119899 sum

119903lt119909minus3119886

119909 + 119899 sum

119903ge119909minus3119886

1

1199031198863]

]

2

+ 119899

119899

sum

119903=1

1199032minus1198863

)

le 1198631sdot (119899

2119909

2(119886minus3)119886+ 120585)

(68)

where119863 and1198631are constants and

120585 = 119899

119899

sum

119903=1

1199032minus1198863

=

119874(119899) 119886 gt 9

119874 (119899 ln 119899) 119886 = 9

119874 (1198994minus1198863

) 3 lt 119886 lt 9

(69)

It is worth mentioning that in the last inequality of (68) weused the estimate

int

infin

119909

1

119905119901119889119905 sim

1

119909119901minus1for 119901 gt 1 (70)

At the same time in the case of MTP2rvrsquos we get

1198641198784

119899le 119863

2sdot (119899

2119909

2(119886minus1)119886+ 120577) (71)

where1198632is constant and

120577 = 119899

119899

sum

119903=1

1199032minus119886

=

119874(119899) 119886 gt 3

119874 (119899 ln 119899) 119886 = 3

119874 (1198994minus119886

) 1 lt 119886 lt 3

(72)

Let now 120572119899(119909) = (1radic119899)sum

119899

119894=1(119868[119880

119905119894

le 119909] minus 119909) Then

120572119899(119909) minus 120572

119899(119910) =

1

radic119899

119899

sum

119894=1

(119868 [119909 lt 119880119905119894

le 119910] minus (119909 minus 119910))

for 119909 119910 isin [0 1]

(73)

8 ISRN Probability and Statistics

has the fourth moment estimatedmdashin the case of associatedrvrsquosmdashby

119864[120572119899(119909) minus 120572

119899(119910)]

4

= 119864[1

radic119899

119899

sum

119894=1

(119868 [119909 lt 119880119905119894

le 119910] minus (119909 minus 119910))]

4

=1

1198992119864[

119899

sum

119894=1

(119868 [119909 lt 119880119905119894

le 119910] minus (119909 minus 119910))]

4

le 1198631sdot (

1

119899211989921003816100381610038161003816119909 minus 119910

1003816100381610038161003816

2(119886minus3)119886

+120585

1198992)

= 1198631sdot (

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

2(119886minus3)119886

+120585

1198992)

=

119874(119899minus1) 119886 gt 9

119874(ln 119899119899

) 119886 = 9

119874 (1198992minus1198863

) 3 lt 119886 lt 9

(74)

and in the case of MTP2rvrsquos by

119864[120572119899(119909) minus 120572

119899(119910)]

4

le 1198632sdot (

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

2(119886minus1)119886

+120577

1198992)

=

119874(119899minus1) 119886 gt 3

119874(ln 119899119899

) 119886 = 3

119874 (1198992minus119886

) 1 lt 119886 lt 3

(75)

In light of the Shao and Yursquos criterion our process is tight forassociated rvrsquos when 119886 gt 6 and for MTP

2rvrsquos when 119886 gt 2

Let us sum up this result in the following theorem

Theorem 3 Let 120572119899(119909) = (1radic119899)sum

119899

119894=1(119868[119880

119894le 119909] minus 119909) be the

empirical process built on an associated sequence of uniformly[0 1] distributed rvrsquos 119880

119894119894ge1

Let also

120579119903= sup

119905isinNCov (119880

119905 119880

119905+119903) = 119874 ((119903 + 1)

minus119886)

where 119903 isin 0 1 119886 gt 0

(76)

Then 120572119899(119909)

119899ge1is tight for 119886 gt 6 If the rvrsquos 119880

119894119894ge1

are MTP2

then the process is tight for 119886 gt 2

Yu assumed stationarity of 119880119894119894ge1

andsum

infin

119899=111989965+]Cov(119880

0 119880

119899) lt infin for a positive constant ]

thus the rate of decay 119886 gt 7 5 Our result weakens consid-erably these assumptions especially in the case of MTP

2rvrsquos

Louhichi in [16] proposed a different tightness criterioninvolving the so-called bracketing numbers She managedto enhance Yursquos resultmdasheven more than Shao and Yu in[8]mdashsince she proved that it suffices to take 119886 gt 4 to gettightness of the empirical process based on the associated

rvrsquos Nevertheless she kept the assumption of stationarityvalid

In the final analysis our resultrsquos advantage is the absenceof the stationarity assumption and the rate of decay for120579119903remains (up to the authorrsquos knowledge) unimproved for

MTP2rvrsquos

Unfortunately with a view to obtainingweak convergenceof the process in question that is also convergence offinite-dimensional distributions we do not know how tomanage without the assumption of stationarityTherefore weconclude with the following corollary

Corollary 4 Let 120572119899(119909) = (1radic119899)sum

119899

119894=1(119868[119880

119894le 119909] minus 119909) be the

empirical process built on a stationary associated sequence ofuniformly [0 1] distributed rvrsquos 119880

119894119894ge1

Let also

120579119903= Cov (119880

1 119880

1+119903) = 119874 ((119903 + 1)

minus119886)

where 119903 isin 0 1 119886 gt 0

(77)

Then if 119886 gt 6

120572119899(sdot) 997888rarr 119861 (sdot) weakly in 119863 [0 1] (78)

where 119861(sdot) is the zero mean Gaussian process on [0 1] withcovariance structure defined by

1205902(119909 119910) = 119909 and 119910 minus 119909119910

+

infin

sum

119895=1

[ Cov (119868 [1198801le 119909] 119868 [119880

119895+1le 119910])

+ Cov (119868 [1198801le 119910] 119868 [119880

119895+1le 119909])]

(79)

If the rvrsquos 119880119894119894ge1

are MTP2 then it suffices to be 119886 gt 2 in

order to claim the above convergence

Proof It remains to establish convergence of finite-dimen-sional distributions repeating the procedure from [7]

4 Tightness of the Kernel-TypeEmpirical Process

In this section we shall weaken assumption imposed on thecovariance structure of rvrsquos 119883

119895119895ge1

by Cai and Roussas in [1]for the kernel estimator of the df

119865119899(119909) =

1

119899

119899

sum

119895=1

119870(

119909 minus 119883119895

ℎ119899

) (80)

They deal with a stationary sequence of negatively associatedrvrsquos (cf [17]) and need the same condition as Yu [7] that is

1003816100381610038161003816Cov (1198831 119883

1+119899)1003816100381610038161003816 = 119874(

1

11989975+]) (81)

to get tightness of the smooth empirical process (see condi-tion (A4) in [1])

ISRN Probability and Statistics 9

It turns out that it suffices to have

1003816100381610038161003816Cov (1198801 119880

1+119899)1003816100381610038161003816 = 119874(

1

1198993119901(119901minus1)) (82)

where 119901 gt 4 is a positive constant taken from the tightnesscriterion (8) It is easy to see that asymptotically we get therate 3

On the way to prove it we will also take use of aRosenthal-type inequality due to Shao andYu (seeTheorem 2in [8]) we shall recall in the following lemma

Lemma 5 Let 119901 gt 2 and 119891 be a real valued function boundedby 1 with bounded first derivative Suppose that 119883

119899119899ge1

is asequence of stationary and associated rvrsquos such that for 119899 isin N

Cov (1198831 119883

119899) = 119874 (119899

minus119887) for some 119887 gt 119901 minus 1 (83)

Then for any 120583 gt 0 there exists some positive constant 119896120583

independent of the function 119891 for which

1198641003816100381610038161003816119878119899

(119891) minus 119864119878119899(119891)

1003816100381610038161003816

119901

le 119896120583(119899

1+12058310038171003817100381710038171003817119891

101584010038171003817100381710038171003817

2

+1198991199012 [

[

119899

sum

119895=1

10038161003816100381610038161003816Cov (119891 (119883

1)119891 (119883

119895))10038161003816100381610038161003816]

]

)

(84)

As we can see the lemma assumes association but itworks for negatively associated rvrsquos as well since in the proofit reaches back the result of Newman (see Proposition 15 in[18]) where both types of association are allowed

Let us recall that 120572119899(119909) = radic119899(119865

119899(119909) minus 119864119865

119899(119909)) where

119865119899(119909) = (1119899)sum

119899

119895=1119870((119909 minus 119883

119895)ℎ

119899) It is easy to see that

1198641003816100381610038161003816120572119899

(119909) minus 120572119899(119910)

1003816100381610038161003816

119901

= 119899minus1199012

11986410038161003816100381610038161003816119878119899(119891 (119883

119895)) minus 119864119878

119899(119891 (119883

119895))10038161003816100381610038161003816

119901

(85)

where

119891 (119883119895) = 119870(

119909 minus 119883119895

ℎ119899

) minus 119870(

119910 minus 119883119895

ℎ119899

)

119878119899(119891) =

119899

sum

119895=1

119891 (119883119895)

(86)

With an intent to use Lemma 5 we need 119891 to be boundedfrom above by 1 (which in our case is obvious) and to havebounded 119891

1015840 thus we assume

sup119905isinR

10038161003816100381610038161003816119870

1015840(119905)

10038161003816100381610038161003816

1

ℎ119899

le119862

119870

ℎ119899

lt infin (87)

Now applying inequality (84) we have

1198641003816100381610038161003816120572119899

(119909) minus 120572119899(119910)

1003816100381610038161003816

119901

le119896120583119899minus1199012

1198991+120583

1198622

119870

ℎ2

119899

+1198991199012[

[

119899

sum

119895=1

10038161003816100381610038161003816Cov (119891 (119883

1)119891 (119883

119895))10038161003816100381610038161003816]

]

1199012

(88)

Newman showed in [18] that

Cov (1198911(119883) 119891

2(119884))

= ∬R2119891

1015840

1(119909) 119891

1015840

2(119910)

times [119875 (119883 le 119909 119884 le 119910) minus 119875 (119883 le 119909) 119875 (119884 le 119910)] 119889119909 119889119910

(89)

if 1198911and 119891

2are real valued functions on R having square

integrable derivatives 1198911015840

1and 119891

1015840

2 respectively and provided

that 1198911(119883) 119891

2(119884) have finite secondmoments In light of that

equation and linearity of covariance we get10038161003816100381610038161003816Cov (119891 (119883

1) 119891 (119883

119895))10038161003816100381610038161003816

=

1003816100381610038161003816100381610038161003816∬

R2[119875 (119883

1lt 119909 minus ℎ

119899119904 119883

119895lt 119909 minus ℎ

119899119905)

minus119875 (1198831lt 119909 minus ℎ

119899119904 119883

119895lt 119910 minus ℎ

119899119905)]

minus [119875 (1198831lt 119909 minus ℎ

119899119904) 119875 (119883

119895lt 119909 minus ℎ

119899119905)

minus119875 (1198831lt 119909 minus ℎ

119899119904) 119875 (119883

119895lt 119910 minus ℎ

119899119905)]

minus [119875 (1198831lt 119910 minus ℎ

119899119904 119883

119895lt 119909 minus ℎ

119899119905)

minus119875 (1198831lt 119910 minus ℎ

119899119904 119883

119895lt 119910 minus ℎ

119899119905)]

minus [119875 (1198831lt 119910 minus ℎ

119899119904) 119875 (119883

119895lt 119910 minus ℎ

119899119905)

minus 119875 (1198831lt 119910 minus ℎ

119899119904)

times 119875 (119883119895lt 119909 minus ℎ

119899119905)] 119896 (119904) 119896 (119905) 119889119904 119889119905

10038161003816100381610038161003816100381610038161003816

(90)

Without the loss of generality we may and do assume that119909 gt 119910 thus10038161003816100381610038161003816Cov (119891 (119883

1) 119891 (119883

119895))10038161003816100381610038161003816

=

1003816100381610038161003816100381610038161003816∬

R2119875 (119910 minus ℎ

119899119904 le 119883

1lt 119909 minus ℎ

119899119904

119910 minus ℎ119899119905 le 119883

119895lt 119909 minus ℎ

119899119905)

minus 119875 (119910 minus ℎ119899119904 le 119883

1lt 119909 minus ℎ

119899119904)

times 119875 (119910 minus ℎ119899119905 le 119883

119895lt 119909 minus ℎ

119899119905) 119896 (119904) 119896 (119905) 119889119904 119889119905

1003816100381610038161003816100381610038161003816

(91)

and by triangle inequality we get10038161003816100381610038161003816Cov (119891 (119883

1) 119891 (119883

119895))10038161003816100381610038161003816

le ∬R2[int

119909minusℎ119899119904

119910minusℎ119899119904

int

119909minusℎ119899119905

119910minusℎ119899119905

1198911198831119883119895(119906 V) 119889119906 119889V

+int

119909minusℎ119899119904

119910minusℎ119899119904

1198911198831(119906) 119889119906int

119909minusℎ119899119905

119910minusℎ119899119905

119891119883119895(V) 119889V] 119896 (119904) 119896 (119905) 119889119904 119889119905

(92)

10 ISRN Probability and Statistics

1198911198831119883119895

(119906 V) 1198911198831

(119906) and 119891119883119895

(V) are the joint pdf of [1198831 119883

119895]

and marginal pdfrsquos of rvrsquos1198831and119883

119895 respectively We need

to further assume that 1198621stands for a common upper bound

of 1198911198831

and 1198911198831119883119895

for all 119895 isin N that is

100381710038171003817100381710038171198911198831

10038171003817100381710038171003817le 119862

1

1003817100381710038171003817100381710038171198911198831119883119895

100381710038171003817100381710038171003817le 119862

1forall119895 ge 1 (93)

Then we finally obtain

10038161003816100381610038161003816Cov (119891 (119883

1) 119891 (119883

119895))10038161003816100381610038161003816le 2119862

1

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

2

where = max 1198621 119862

2

1

(94)

On the other hand proceeding like Cai and Roussas in [1] wecan shortly get

10038161003816100381610038161003816Cov (119891 (119883

1) 119891 (119883

119895))10038161003816100381610038161003816

le

100381610038161003816100381610038161003816100381610038161003816

Cov(119870(119909 minus 119883

1

ℎ119899

) 119870(

119909 minus 119883119895

ℎ119899

))

100381610038161003816100381610038161003816100381610038161003816

+

100381610038161003816100381610038161003816100381610038161003816

Cov(119870(119909 minus 119883

1

ℎ119899

) 119870(

119910 minus 119883119895

ℎ119899

))

100381610038161003816100381610038161003816100381610038161003816

+

100381610038161003816100381610038161003816100381610038161003816

Cov(119870(119910 minus 119883

1

ℎ119899

) 119870(

119909 minus 119883119895

ℎ119899

))

100381610038161003816100381610038161003816100381610038161003816

+

100381610038161003816100381610038161003816100381610038161003816

Cov(119870(119910 minus 119883

1

ℎ119899

) 119870(

119910 minus 119883119895

ℎ119899

))

100381610038161003816100381610038161003816100381610038161003816

le 41198622

10038161003816100381610038161003816Cov (119883

1 119883

119895)10038161003816100381610038161003816

13

(95)

where 1198622is a constant relevant to the covariance inequality

for negatively associated rvrsquos (see [13])We now arrive at the following inequality

10038161003816100381610038161003816Cov (119891 (119883

1) 119891 (119883

119895))10038161003816100381610038161003816

le max 1198621 119862

2

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

2

10038161003816100381610038161003816Cov(119883

1 119883

119895)10038161003816100381610038161003816

13

(96)

Assuming |Cov(119891(1198831) 119891(119883

119895))| = 119874(119895

minus119903) and proceeding

similarly to (68) we can get

[

[

119899

sum

119895=1

10038161003816100381610038161003816Cov (119891 (119883

1) 119891 (119883

119895))10038161003816100381610038161003816]

]

1199012

le max 1198621 119862

2 [

[

sum

119895lt|119909minus119910|minus2(1199033)

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

2

+ sum

119895ge|119909minus119910|minus2(1199033)

1

1198951199033]

]

1199012

le 2max 1198621 119862

21003816100381610038161003816119909 minus 119910

1003816100381610038161003816

((119903minus3)119903)119901

(97)

To sum up we obtain the following inequality

1198641003816100381610038161003816120572119899

(119909) minus 120572119899(119910)

1003816100381610038161003816

119901

le 119896120583119899

1+120583minus1199012119862

2

119870

ℎ2

119899

+ 2max 1198621 119862

21003816100381610038161003816119909 minus 119910

1003816100381610038161003816

((119903minus3)119903)119901

le 1198621198991+120583minus1199012 1

ℎ2

119899

+1003816100381610038161003816119909 minus 119910

1003816100381610038161003816

((119903minus3)119903)119901

(98)

where 119862 = 119896120583max1198622

119896 2max119862

1 119862

2 The formula of the

tightness criterion (8) implies that ((119903 minus 3)119903)119901 gt 1 so

119903 gt3119901

119901 minus 1 where 119901 gt 2 (99)

Let us recall the assumptions imposed on the bandwidhtsℎ

119899119899ge1

by Cai and Roussas in [1]B1 lim

119899rarrinfinℎ119899= 0 and ℎ

119899gt 0 for all 119899 isin N

+

B2 lim119899rarrinfin

119899ℎ119899= infin thus ℎ

119899= 119874(1119899

1minus120573) 120573 gt 0

B3 lim119899rarrinfin

119899ℎ4

119899= 0 hence ℎ

119899= 119874(1119899

14+120575) 120575 gt 0

In light of the above

1198991+120583minus1199012 1

ℎ2

119899

= 11989932+120583+2120575minus1199012

(100)

where 120583 gt 0 119901 gt 2 and 0 lt 120575 lt 34 Confrontation with thetightness criterion (8) forces

3

2+ 120583 + 2120575 minus

119901

2lt minus

1

2 (101)

which implies 119901 gt 4 Let us conclude with the followingtheorem

Theorem6 Let 120572119899(119909) = radic119899(119865

119899(119909)minus119864119865

119899(119909)) be the empirical

process built on the kernel estimator of the df119865 for a stationarysequence of negatively associated rvrsquos Assume that conditionsA1 A2 (87) B1 B2 B3 are satisfied Then provided that thetightness criterion (8) holds with 119901 gt 4 it suffices to demand

1003816100381610038161003816Cov (1198801 119880

1+119899)1003816100381610038161003816 = 119874(

1

1198993119901(119901minus1)) (102)

in order to obtain120572

119899(sdot) 997888rarr 119861 (sdot) weakly in 119863 [0 1] (103)

where 119861(sdot) is the zero mean Gaussian process with covariancestructure defined by

1205902(119909 119910) = 119865 (119909 and 119910) minus 119865 (119909) 119865 (119910)

+

infin

sum

119895=1

[ Cov (119868 [1198831le 119909] 119868 [119883

119895+1le 119910])

+ Cov (119868 [1198831le 119910] 119868 [119883

119895+1le 119909])]

(104)

and 119909 119910 isin [0 1]

Proof The remaining covergence of finite-dimensional dis-tributions of 120572

119899(119909) is established in [1]

ISRN Probability and Statistics 11

5 Weak Convergence of theRecursive Kernel-Type Empirical Processunder IID Assumption

The aim of this section is to show weak convergence of theempirical process

120572119899(119909) =

1

radic119899

119899

sum

119895=1

[119870(

119909 minus 119883119895

ℎ119895

) minus 119864119870(

119909 minus 119883119895

ℎ119895

)] (105)

built on iid rvrsquos 119883119895119895ge1

Wewill prove that120572

119899(sdot) convergesweakly to the Brownian

bridge 119861(sdot) with the following covariance structure

1205902(119909 119910) = 119865 (119909 and 119910) minus 119865 (119909) 119865 (119910) for 119909 119910 isin [0 1]

(106)

It turns out that the tightness criterion (8) suffices to reachthe goal Convergence of finite-dimensional distributions of120572

119899(sdot) holds and we shall show itProceeding like Cai and Roussas in [1] we need to show

that for any 119886 119887 isin R

119886120572119899(119909) + 119887120572

119899(119910) 997888rarr 119886119861 (119909) + 119887119861 (119910) in distribution

(107)

Let us introduce the notation

119884119894(119909) = 119870(

119909 minus 119883119894

ℎ119894

) minus 119864119870(119909 minus 119883

119894

ℎ119894

) (108)

and look into the covariance structure of 120572119899(sdot)

Cov (120572119899(119909) 120572

119899(119910))

= Cov( 1

radic119899

119899

sum

119894=1

119884119894(119909)

1

radic119899

119899

sum

119894=1

119884119894(119910))

=1

119899

119899

sum

119894=1

Cov (119884119894(119909) 119884

119894(119910))

=1

119899

119899

sum

119894=1

Cov(119870(119909 minus 119883

119894

ℎ119894

) 119870(119910 minus 119883

119894

ℎ119894

))

(109)

where the second equality follows from assumed indepen-dence of rvrsquos 119883

119894119894ge1

Firstly we observe that each summand converges to119865(119909and

119910) minus 119865(119909)119865(119910) since

Cov(119870(119909 minus 119883

119894

ℎ119894

) 119870(119910 minus 119883

119894

ℎ119894

))

= 119864[119870(119909 minus 119883

119894

ℎ119894

)119870(119910 minus 119883

119894

ℎ119894

)]

minus 119864119870(119909 minus 119883

119894

ℎ119894

)119864119870(119910 minus 119883

119894

ℎ119894

)

(110)

where

119864[119870(119909 minus 119883

119894

ℎ119894

)119870(119910 minus 119883

119894

ℎ119894

)]

= int

119909and119910

minusinfin

119870(119909 minus 119905

ℎ119894

)119870(119910 minus 119905

ℎ119894

)119889119865 (119905)

+ int

119909or119910

119909and119910

119870(119909 minus 119905

ℎ119894

)119870(119910 minus 119905

ℎ119894

)119889119865 (119905)

+ int

infin

119909or119910

119870(119909 minus 119905

ℎ119894

)119870(119910 minus 119905

ℎ119894

)119889119865 (119905)

(111)

and 119865 denotes the common df of the rvrsquos 119883119894119894ge1

In [6] itwas shown that

119864119870(119909 minus 119883

119894

ℎ119894

) 997888rarr 119865 (119909) (112)

and recalling that the kernel function 119870 is a df as well wearrive at the conclusion

Secondly applying Toeplitz lemma we get

Cov (120572119899(119909) 120572

119899(119910)) 997888rarr 119865 (119909 and 119910) minus 119865 (119909) 119865 (119910)

= 1205902(119909 119910) 119899 997888rarr infin

(113)

Thus

Var (119886120572119899(119909) + 119887120572

119899(119910))

= 1198862 Var (120572

119899(119909)) + 2119886119887Cov (120572

119899(119909) 120572

119899(119910))

+ 1198872 Var (120572

119899(119910))

997888rarr 1198862120590

2(119909 119909) + 2119886119887120590

2(119909 119910) + 119887

2120590

2(119910 119910) 119899 997888rarr infin

(114)

We are now on the way to prove that 119886120572119899(119909) + 119887120572

119899(119910) con-

verges in distribution to 119886119861(119909) + 119887119861(119910) sim N(0 1198862120590

2(119909 119909) +

21198861198871205902(119909 119910) + 119887

2120590

2(119910 119910)) which in other words means that

1

radic119899

infin

sum

119894=1

119886119884119894(119909) + 119887119884

119894(119910)

radicVar (119886120572119899(119909) + 119887120572

119899(119910))

997888rarr N (0 1) (115)

In order to obtain (115) we will use Lyapunov condition

lim119899rarrinfin

1

1198632+120575

119899

119899

sum

119894=1

1198641003816100381610038161003816119886119884119894

(119909) + 119887119884119894(119910)

1003816100381610038161003816

2+120575

= 0 (116)

for 120575 = 1 where 1198632

119899= Var(sum119899

119894=1[119886119884

119894(119909) + 119887119884

119894(119910)]) Since

|119870(sdot) minus 119864119870(sdot)| le 2

1198641003816100381610038161003816119886119884119894

(119909) + 119887119884i (119910)1003816100381610038161003816

3

le 8 (1198863+ 119887

3) (117)

Moreover 1198632

119899= 119899Var(119886120572

119899(119909) + 119887120572

119899(119910)) together with (114)

yields

1198633

119899= 119874 (119899

32) (118)

12 ISRN Probability and Statistics

Combining (117) with (118) we obtain

1

1198633

119899

119899

sum

119894=1

1198641003816100381610038161003816119886119884119894

(119909) + 119887119884119894(119910)

1003816100381610038161003816

3

= 119874 (119899minus12

) (119)

which shows that Lyapunov condition holds and completesthe proof of convergence of finite-dimensional distributionsof 120572

119899(119909)We summarize the result of that section in the following

theorem

Theorem 7 Let 120572119899(119909) be the recursive kernel-type empirical

process defined by (105) built on iid rvrsquos 119883119895119895ge1

withmarginal df 119865 If 120572

119899(sdot) satisfies the tightness criterion (8) then

120572119899(sdot) 997888rarr 119861 (sdot) weakly in 119863 [0 1] (120)

where 119861 is the Brownian bridge

Acknowledgment

The author would like to thank the referees for the carefulreading of the paper and helpful remarks

References

[1] Z Cai and G G Roussas ldquoWeak convergence for smoothestimator of a distribution function under negative associationrdquoStochastic Analysis and Applications vol 17 no 2 pp 145ndash1681999

[2] Z Cai andG G Roussas ldquoEfficient Estimation of a DistributionFunction under Qudrant Dependencerdquo Scandinavian Journal ofStatistics vol 25 no 1 pp 211ndash224 1998

[3] ZW Cai and G G Roussas ldquoUniform strong estimation under120572-mixing with ratesrdquo Statistics amp Probability Letters vol 15 no1 pp 47ndash55 1992

[4] G G Roussas ldquoKernel estimates under association stronguniform consistencyrdquo Statistics amp Probability Letters vol 12 no5 pp 393ndash403 1991

[5] Y F Li and S C Yang ldquoUniformly asymptotic normalityof the smooth estimation of the distribution function underassociated samplesrdquo Acta Mathematicae Applicatae Sinica vol28 no 4 pp 639ndash651 2005

[6] Y Li C Wei and S Yang ldquoThe recursive kernel distributionfunction estimator based onnegatively and positively associatedsequencesrdquo Communications in Statistics vol 39 no 20 pp3585ndash3595 2010

[7] H Yu ldquoA Glivenko-Cantelli lemma and weak convergence forempirical processes of associated sequencesrdquo ProbabilityTheoryand Related Fields vol 95 no 3 pp 357ndash370 1993

[8] Q M Shao and H Yu ldquoWeak convergence for weightedempirical processes of dependent sequencesrdquo The Annals ofProbability vol 24 no 4 pp 2098ndash2127 1996

[9] P Billingsley Convergence of Probability Measures Wiley Seriesin Probability and Statistics Probability and Statistics JohnWiley amp Sons New York NY USA 1999

[10] J D Esary F Proschan and D W Walkup ldquoAssociation ofrandom variables with applicationsrdquo Annals of MathematicalStatistics vol 38 pp 1466ndash1474 1967

[11] A Bulinski and A Shashkin Limit Theorems for AssociatedRandom Fields and Related Systems vol 10 of Advanced Serieson Statistical Science and Applied Probability World Scientific2007

[12] S Karlin and Y Rinott ldquoClasses of orderings of measures andrelated correlation inequalities I Multivariate totally positivedistributionsrdquo Journal of Multivariate Analysis vol 10 no 4 pp467ndash498 1980

[13] P Matuła and M Ziemba ldquoGeneralized covariance inequali-tiesrdquo Central European Journal of Mathematics vol 9 no 2 pp281ndash293 2011

[14] P Doukhan and S Louhichi ldquoA new weak dependence condi-tion and applications to moment inequalitiesrdquo Stochastic Proc-esses and their Applications vol 84 no 2 pp 313ndash342 1999

[15] P Matuła ldquoA note on some inequalities for certain classes ofpositively dependent random variablesrdquo Probability and Math-ematical Statistics vol 24 no 1 pp 17ndash26 2004

[16] S Louhichi ldquoWeak convergence for empirical processes of asso-ciated sequencesrdquo Annales de lrsquoInstitut Henri Poincare vol 36no 5 pp 547ndash567 2000

[17] K Joag-Dev and F Proschan ldquoNegative association of randomvariables with applicationsrdquoThe Annals of Statistics vol 11 no1 pp 286ndash295 1983

[18] C M Newman ldquoAsymptotic independence and limit theoremsfor positively and negatively dependent random variablesrdquo inInequalities in Statistics and Probability Y L Tong Ed vol 5pp 127ndash140 Institute ofMathematical StatisticsHayward CalifUSA 1984

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 ISRN Probability and Statistics

In this paper we shall study the empirical process 120572119899(119909)

generated by the generalized kernel estimator of the df givenby the formula

119865119899(119909) =

1

119899

119899

sum

119895=1

119870(

119909 minus 119883119895

ℎ119899119895

) where (5)

A1 119883119895119895ge1

is a sequence of absolutely continuous id rvrsquostaking values in [0 1] and having twice differentiabledf 119865 with first and second derivative bounded

A2 119870 is a kernel function such that intR 119906119889119870(119906) = 0 andintR 119906

2119889119870(119906) lt infin with bounded derivative 119896

A3 ℎ119899119895119899ge1119895isin1119899

is a sequence of positive constantssubject to the following conditions lim

119895119899rarrinfinℎ119899119895

= 0lim

119895119899rarrinfin119899ℎ

4

119899119895= 0 (actually since 119895 le 119899 119895 rarr infin

under the limit suffices)

Explicitly we take a look onto the process

120572119899(119909) =

1

radic119899sdot

119899

sum

119895=1

[119870(

119909 minus 119883119895

ℎ119899119895

) minus 119864119870(

119909 minus 119883119895

ℎ119899119895

)] (6)

we shall call from now on the generalized empirical processLet us pay attention to the fact that in the case of

(i) ℎ119899119895

= ℎ119899for 119895 isin 1 119899 120572

119899(119909) is the empirical

process based on the kernel estimator of the df 119865(ii) ℎ

119895119895= ℎ

119895+1119895= ℎ

119895+2119895= sdot sdot sdot = ℎ

119895for 119895 isin N 120572

119899(119909)

is the empirical process based on the recursive kernelestimator of the df 119865

(iii) 119870(119905) = 119868[0infin]

(119905) 120572119899(119909) is the standard empirical

process (based on the empirical df)

It is well known that the crucial procedure in showingweak convergence for an empirical process is to verify tight-ness In [8] Shao and Yu gave the following criterion

exist1198621gt0exist

119901gt2exist

1199011gt1exist

0le1199031le1exist

1199012gt1minus1199031

forall119909119910isin[01]

1198641003816100381610038161003816120572119899

(119909) minus 120572119899(119910)

1003816100381610038161003816

119901

le 1198621(1003816100381610038161003816119909 minus 119910

1003816100381610038161003816

1199011

+ 119899minus119901221003816100381610038161003816119909 minus 119910

1003816100381610038161003816

1199031

)

(7)

under which the standard empirical process based on sta-tionary sequence of uniform [0 1] rvrsquos is tight It is statedthere that the proof of that fact is an easy standard procedureparallel to the one presented in [9] It is the main aim of thispaper to carry it in details but for the generalized empiricalprocess defined by (6) and without assuming stationarityNevertheless wewill always return to stationarity assumptionwhile establishing weak convergence

In order to obtain tightness one has to assume appro-priate covariance structure of the underlying rvrsquos that isthe covariance of a pair of rvrsquos 119883

119894and 119883

119895has to decline

at the right rate while 119894 and 119895 are growing apart In thispaper we lower the demanded rate of covariance decay usingthe covariance inequalities for associated (cf [10 11]) andmultivariate totally positive of order 2 (MTP

2) (cf [12]) rvrsquos

obtained in [13]

The paper is organized as follows In Section 2 we presentthe proof of the Shao and Yursquos tightness criterion formulatedfor our generalized empirical process Sections 3 and 4 aredevoted to application of the criterion to showing tightnessand thus weak convergence of the specific types of empiricalprocesses Section 5 concerns weak convergence of the recur-sive kernel-type process for iid rvrsquos

2 Tightness Criterion

We start with the key point of the paper

Theorem 1 Let 120572119899(119909)

119899ge1isin 119863[0 1] be the generalized empir-

ical process defined as in (6) One assume that A1 A2 A3 holdIf there exist constants 119862 gt 0 119901 gt 2 119901

1gt 1 0 le 119901

3le 1

1199012gt 1 minus 119901

3 such that for any 119909 119910 isin [0 1] and 119899 isin N the

following inequality holds

1198641003816100381610038161003816120572119899

(119909) minus 120572119899(119910)

1003816100381610038161003816

119901

le 119862 sdot (1003816100381610038161003816119909 minus 119910

1003816100381610038161003816

1199011

+ 119899minus11990122sdot1003816100381610038161003816119909 minus 119910

1003816100381610038161003816

1199013

)

(8)

then the process 120572119899(119909)

119899ge1is tight in119863[0 1] with 119869

1topology

Proof The proof boils down to showing that under theassumptions made inTheorem 1 conditions ofTheorem 132in [9] hold Let us recall that in light of the above mentionedtheorem a process 120572

119899(119909)

119899ge1is tight in119863[0 1] if

lim119886rarrinfin

lim sup119899rarrinfin

119875 (1003817100381710038171003817120572119899

1003817100381710038171003817 ge 119886) = 0 (9)

lim120575rarr0

lim sup119899rarrinfin

119875 (1199081015840

1(120572

119899 120575) ge 120598) = 0 forall120598 gt 0 (10)

where sdot is the supremum norm that is1003817100381710038171003817120572119899

1003817100381710038171003817 = sup0le119909le1

1003816100381610038161003816120572119899(119909)

1003816100381610038161003816 (11)

and 1199081015840

1(120572

119899 120575) is the modulus of continuity of the function

120572119899isin 119863[0 1] that is

1199081015840

1(120572

119899 120575) = inf

119905119894

max1le119894le]

sup119909119910isin[119905119894minus1119905119894)

1003816100381610038161003816120572119899(119909) minus 120572

119899(119910)

1003816100381610038161003816 (12)

The infimum runs over all finite ldquo120575-sparserdquo decompositions[119905

119894minus1 119905

119894) 119894 isin 1 ] of the interval [0 1] In other words it

runs over all choices of increasingly ordered points 1199051198941le119894le]

such that min1le119894le](119905119894 minus 119905

119894minus1) gt 120575 where 119905

0= 0 119905] = 1

Let us first show that condition (9) is satisfied Accordingto the corollary following Theorem 132 in [9] it suffices toshow

lim119886rarrinfin

lim sup119899rarrinfin

119875 (1205721198991003816100381610038161003816120572119899

(119909)1003816100381610038161003816 ge 119886) = 0 forall119909 isin [0 1] (13)

that is

forall120578ge0

exist119886gt0

forall119899isinN119875 (120572

1198991003816100381610038161003816120572119899

(119909)1003816100381610038161003816 ge 119886) le 120578 forall119909 isin [0 1] (14)

Let us fix 119909 isin [0 1] and 120578 gt 0 We need to find 119886 = 119886(120578) gt 0such that

119875 (radic11989910038161003816100381610038161003816119865119899(119909) minus 119864119865

119899(119909)

10038161003816100381610038161003816ge 119886) le 120578 forall119899 isin N (15)

ISRN Probability and Statistics 3

where

119865119899(119909) =

1

119899

119899

sum

119895=1

119870(

119909 minus 119883119895

ℎ119899119895

) (16)

is the generalized kernel estimator of the df119865 ApplyingChe-byshevrsquos inequality we shall find 119886 satisfying

119886 geradic119899

120578sdot 119864

10038161003816100381610038161003816119865119899(119909) minus 119864119865

119899(119909)

10038161003816100381610038161003816forall119899 isin N (17)

Such 119886 exists if 119864|120572119899(119909)| lt infin for all 119899 ge 1 Implementing

119910 = 0 and 1199013= 0 in the assumption (8) we get for 119901 gt 2

1199011 119901

2gt 1

1198641003816100381610038161003816120572119899

(119909)1003816100381610038161003816

119901

le 119862 sdot(|119909|1199011+119899

minus11990122|119909|

1199013) le 119862 sdot(1+

1

11989911990122) le 2119862

(18)

Now applying Holderrsquos inequality we arrive at

1198641003816100381610038161003816120572119899

(119909)1003816100381610038161003816 le

4radic119864

1003816100381610038161003816120572119899(119909)

1003816100381610038161003816

4

(19)

which in light of the inequality (18) for 119901 = 4 is boundedfrom above by 4radic2119862 lt infin Therefore condition (9) holds

We shall now proceed to checking condition (10) Let usrecall that the modulus of continuity of the function 119909 isin

119862[0 1] 120575 gt 0 is given by the formula

1199081(119909 120575) = sup

0le119910le1minus120575

sup119910le119909le119910+120575

1003816100381610038161003816120572119899(119909) minus 120572

119899(119910)

1003816100381610038161003816 (20)

As it is shown in [9] (page 123) there is a relation between119908

1(sdot sdot) and 1199081015840

1(sdot sdot) in the spaces 119862[0 1] and119863[0 1] relatively

Namely for 119909 isin 119863[0 1] and 120575 lt 12

1199081(119909 2120575) ge 119908

1015840

1(119909 120575) (21)

Thus

119875 (1199081015840

1(120572

119899 120575) ge 120598) le 119875 (119908

1(120572

119899 2120575) ge 120598) (22)

Therefore condition (10) holds if we show

forall120598gt0

forall120578gt0

exist0le120575lt12

exist1198990

forall119899ge1198990

119875 (1199081(120572

119899 2120575) ge 120598) le 120578 (23)

With a view to obtaining the desired inequality we shall pro-ceed patiently in five steps

Step 1 We need a moment inequality for the rv 120572119899(119909)minus120572

119899(119910)

involving the distance between the points 119909 and 119910 Let usthen assume that for the constants 119862 119901 119901

1 119901

2 119901

3given in

Theorem 1 inequality (8) holds Fix 120598 gt 0 120578 gt 0 and definethe quantity 119903

119899= 120598radic119899 119899 isin N Next we fix 119909 119910 isin [0 1] and

take 119899 large enough so that 119903119899le |119909minus119910|Then 119899minus12

le |119909minus119910|120598minus1

and we have

1198641003816100381610038161003816120572119899

(119909) minus 120572119899(119910)

1003816100381610038161003816

119901

le 119862 (1 + 120598minus1199012)1003816100381610038161003816119909 minus 119910

1003816100381610038161003816

min11990111199012+1199013

(24)

Step 2 Let us now fix 119899 isin N 120575 isin (0 1) and consider the fol-lowing rvrsquos

120594119894= 120572

119899(119910 + 119894119903

119899) minus 120572

119899(119910 + (119894 minus 1) 119903

119899) (25)

for 119894 isin 1 2 119898119899 where119898

119899= 119898(119899 120575) is such that 119903

119899119898

119899le 120575

It is easy to see that for 119878119894= sum

119894

119896=1120594

119896

max1le119894le119898

119899

1003816100381610038161003816119878119894

1003816100381610038161003816 = max1le119894le119898

119899

1003816100381610038161003816120572119899(119910 + 119894119903

119899) minus 120572

119899(119910)

1003816100381610038161003816 (26)

Let us notice that for rvrsquos 120594119894119894ge1

the conditions of Theo-rem 102 in [9] are satisfied with 4120573 = 119901 119906

119897= 119903

119899forall

119894lt119897le119895and

2120572 = min1199011 119901

2+ 119901

3 Therefore we are equipped with the

following maximal inequality

119875( max1le119894le119898

119899

1003816100381610038161003816120572119899(119910 + 119894119903

119899) minus 120572

119899(119910)

1003816100381610038161003816 ge 120582)

le

119862119901min119901

11199012+1199013

120582119901(119898

119899119903119899)min119901

11199012+1199013

(27)

for all 120582 gt 0

Step 3 Let 119872 = sup119909isin[01]

119891(119909) where 119891 is the pdf of rvrsquos119883

119895119895ge1

For fixed 120598 120578 gt 0 let us take 120575 gt 0 such that

119862119901min119901

11199012+1199013

(119872120598)119901

sdot (2120575)min119901

11199012+1199013lt 120578 (28)

and define119898119899= lfloor120575119903

119899rfloor For sufficiently large 119899 isin N we have

119898119899ge 1 and then we get

119903119899119898

119899le 120575 lt (119898

119899+ 1) 119903

119899le 2119898

119899119903119899le 120575 (29)

Step 4 Our goal is to obtain an inequality which enables usto bound the supremum of the increment of the function 120572

119899

via the maximum of the increments of that function on somesubintervals To bemore precise wewill find the upper boundfor

sup119910le119909le119910+119898

119899119902

1003816100381610038161003816120572119899(119909) minus 120572

119899(119910)

1003816100381610038161003816 (30)

in terms of

max1le119894le119898

119899

1003816100381610038161003816120572119899(119910 + 119894119902) minus 120572

119899(119910)

1003816100381610038161003816 (31)

where 119909 119910 isin [0 1] 119910 le 119909 le 119910 + 119902 119902 ge 120598119899 and119898119899is defined

as in Step 3Let us recall that 120572

119899(119909) = radic119899(119865

119899(119909) minus 119864119865

119899(119909)) where

119865119899(119909) = (1119899)sum

119899

119895=1119870((119909 minus 119883

119895)ℎ

119899119895) From the triangle ine-

quality we can see that

1003816100381610038161003816120572119899(119909) minus 120572

119899(119910)

1003816100381610038161003816

leradic119899

119899

10038161003816100381610038161003816100381610038161003816100381610038161003816

119899

sum

119895=1

[119870(

119909 minus 119883119895

ℎ119899119895

) minus 119870(

119910 minus 119883119895

ℎ119899119895

)]

10038161003816100381610038161003816100381610038161003816100381610038161003816

+ radic119899 (10038161003816100381610038161003816119864119865

119899(119909) minus 119865 (119909)

10038161003816100381610038161003816+10038161003816100381610038161003816119864119865

119899(119910) minus 119865 (119910)

10038161003816100381610038161003816

+1003816100381610038161003816119865 (119909) minus 119865 (119910)

1003816100381610038161003816 )

(32)

4 ISRN Probability and Statistics

Since 119870 is a nondecreasing function and applying triangleinequality again we get

1003816100381610038161003816120572119899(119909) minus 120572

119899(119910)

1003816100381610038161003816

le1003816100381610038161003816120572119899

(119910 + 119902) minus 120572119899(119910)

1003816100381610038161003816 +radic119899

10038161003816100381610038161003816119864119865

119899(119910 + 119902) minus 119864119865

119899(119910)

10038161003816100381610038161003816

+ radic119899 (10038161003816100381610038161003816119864119865

119899(119909) minus 119865 (119909)

10038161003816100381610038161003816+10038161003816100381610038161003816119864119865

119899(119910) minus 119865 (119910)

10038161003816100381610038161003816

+1003816100381610038161003816119865 (119909) minus 119865 (119910)

1003816100381610038161003816 )

(33)

Cai and Roussas in [1] showed that under assumptions madeon the df 119865 and the kernel function119870 we have

10038161003816100381610038161003816119864119865

119899(119909) minus 119865 (119909)

10038161003816100381610038161003816= 119874 (ℎ

2

119899) (34)

Similarly by Taylor expansion it is easy to see that

1003816100381610038161003816119865 (119909) minus 119865 (119910)1003816100381610038161003816 le

10038171003817100381710038171198911003817100381710038171003817 sdot

1003816100381610038161003816119909 minus 1199101003816100381610038161003816 (35)

Thus

1003816100381610038161003816120572119899(119909) minus 120572

119899(119910)

1003816100381610038161003816 le1003816100381610038161003816120572119899

(119910 + 119902) minus 120572119899(119910)

1003816100381610038161003816

+ 119862radic119899ℎ4

119899+119872radic119899119902

(36)

where 119872 = sup119909isin[01]

119891(119909) and 119862 = 1198621198911015840119870

is a positive con-stant dependent on the functions 119865 and119870

Let us recall that 119910 le 119909 le 119910+ 119902119898119899= lfloor120575119903

119899rfloor and 119902 ge 120598119899

Taking 119902 = (120598radic119899)(= 119903119899) we notice that 119898

119899119902 = lfloor120575119903

119899rfloor119903

119899le 120575

thus when 119899 rarr infin the interval [119910 119910 + 119898119899119902] coincides with

[119910 119910 + 120575] Approaching the aim of Step 4 let us observe that

sup119910le119909le119910+119898

119899119902

1003816100381610038161003816120572119899(119909) minus 120572

119899(119910)

1003816100381610038161003816 = max1le119894le119898

119899

1003816100381610038161003816120572119899(119910 + 119894119902) minus 120572

119899(119910)

1003816100381610038161003816

+1003816100381610038161003816120572119899

(1199090) minus 120572

119899(119910 + 119894max119902)

1003816100381610038161003816

(37)

where 1199090isin [119910+119894

0119902 119910+(119894

0+1)119902] is a point at which the above

supremum is attained and

119894max =

1198940+ 1 if 1003816100381610038161003816120572119899

(119910 + [1198940+ 1] 119902) minus 120572

119899(119910)

1003816100381610038161003816

ge1003816100381610038161003816120572119899

(119910 + 1198940119902) minus 120572

119899(119910)

1003816100381610038161003816

1198940

otherwise(38)

Without the loss of generality we may assume that 119894max = 1198940+

1 which implies

sup119910le119909le119910+119898

119899119902

1003816100381610038161003816120572119899(119909) minus 120572

119899(119910)

1003816100381610038161003816 = max1le119894le119898

119899

1003816100381610038161003816120572119899(119910 + 119894119902) minus 120572

119899(119910)

1003816100381610038161003816

+1003816100381610038161003816120572119899

(1199090) minus 120572

119899(119910 + 119894

0119902)1003816100381610038161003816

(39)

Now applying inequality (36) the definition of 119894max and thetriangle inequality we have

sup119910le119909le119910+119898

119899119902

1003816100381610038161003816120572119899(119909) minus 120572

119899(119910)

1003816100381610038161003816

le max1le119894le119898

119899

1003816100381610038161003816120572119899(119910 + 119894119902) minus 120572

119899(119910)

1003816100381610038161003816

+1003816100381610038161003816120572119899

(119910 + [1198940+ 1] 119902) minus 120572

119899(119910 + 119894

0119902)1003816100381610038161003816

+ 119872radic119899119902 + 119862radic119899ℎ4

119899

le max1le119894le119898

119899

1003816100381610038161003816120572119899(119910 + 119894119902) minus 120572

119899(119910)

1003816100381610038161003816

+1003816100381610038161003816120572119899

(119910 + [1198940+ 1] 119902) minus 120572

119899(119910)

1003816100381610038161003816

+1003816100381610038161003816120572119899

(119910 + 1198940119902) minus 120572

119899(119910)

1003816100381610038161003816 + 119872radic119899119902 + 119862radic119899ℎ4

119899

le 3 max1le119894le119898

119899

1003816100381610038161003816120572119899(119910 + 119894119902) minus 120572

119899(119910)

1003816100381610038161003816 + 119872radic119899119902 + 119862radic119899ℎ4

119899

(40)

If we plug in 119902 = 119903119899we arrive at

sup119910le119909le119910+119898

119899119903119899

1003816100381610038161003816120572119899(119909) minus 120572

119899(119910)

1003816100381610038161003816 le 3 max1le119894le119898

119899

1003816100381610038161003816120572119899(119910 + 119894119903

119899) minus 120572

119899(119910)

1003816100381610038161003816

+ 119872radic119899119903119899+ 119862radic119899ℎ4

119899

(41)

Step 5 Finally we are in a position to obtain inequality (23)that is

forall120598gt0forall

120578gt0exist

0le120575lt12exist

1198990

forall119899ge1198990

119875( sup0le119910le1minus2120575

sup119910le119909le119910+2120575

1003816100381610038161003816120572119899(119909)minus120572

119899(119910)

1003816100381610038161003816

ge 120598) le 120578

(42)

We now successivelymake use the inequalities (29) (41) (27)(29) and (28) to get

119875( sup119910le119909le119910+2120575

1003816100381610038161003816120572119899(119909) minus 120572

119899(119910)

1003816100381610038161003816 ge 4119872120598 + 119862radic119899ℎ4

119899)

le 119875( sup119910le119909le119910+2(119898

119899+1)119903119899

1003816100381610038161003816120572119899(119909) minus 120572

119899(119910)

1003816100381610038161003816 ge 4119872120598 + 119862radic119899ℎ4

119899)

le 119875(3 max1le119894le2(119898119899+1)

1003816100381610038161003816120572119899(119910 + 119894119903

119899) minus 120572

119899(119910)

1003816100381610038161003816

+119872radic119899119903119899+ 119862radic119899ℎ4

119899ge 4119872120598 + 119862radic119899ℎ4

119899)

ISRN Probability and Statistics 5

= 119875( max1le119894le2(119898

119899+1)

1003816100381610038161003816120572119899(119910 + 119894119903

119899) minus 120572

119899(119910)

1003816100381610038161003816 ge 119872120598)

le

119862119901min119901

11199012+1199013

(119872120598)119901

(2120575)min119901

11199012+1199013le 120578

(43)

for any fixed 119910 isin [0 1] Since the upper bound does notdepend on 119910 and the probability measure as well as supre-mum function are continuous we obtain

119875( sup0le119910le1minus2120575

sup119910le119909le119910+2120575

1003816100381610038161003816120572119899(119909) minus 120572

119899(119910)

1003816100381610038161003816 ge 4119872120598 + 119862radic119899ℎ4

119899) le 120578

(44)

where 4119872120598 + 119862radic119899ℎ4

119899is arbitrarily small

Since condition (10) is checked the proof is completed

3 Tightness of the Standard Empirical Process

In this section we deal with the standard empirical processbuilt on an associated sequence of uniformly [0 1]distributedrvrsquos 119880

119895119895ge1

that is

120572119899(119909) =

1

radic119899

119899

sum

119895=1

(119868 [119880119895le 119909] minus 119909) (45)

where 119909 isin [0 1] We shall relax the restrictions imposedon the process by Yu in [7] to obtain tightness Preciselywe do not need stationarity any more due to the techniquedrawn from [14] and we lower the assumed rate at which thecovariance tends to zero

While proving tightness of the empirical process we willuse the criterion proved in the first section as well as some ofour covariance inequalities

We shall start with the fact known under the name ofmultinomial theorem We recall it in the following lemma

Lemma 2 For natural numbers119898 119899 and 1199091 119909

2 119909

119898isin R

(1199091+ sdot sdot sdot + 119909

119899)119898

= sum

1198961+sdotsdotsdot+119896

119899=119898

(119898

1198961 119896

119899

) prod

1le119894le119899

119909119896119894

119894 (46)

where (119898

1198961 119896

119899) = 119898(119896

1 sdot sdot sdot 119896

119899) and 119896

1 119896

119899isin

0 1 119898

In particular

(1199091+ sdot sdot sdot + 119909

119899)4

le 4 sdot sum

1198961+sdotsdotsdot+119896

119899=4

1199091198961

1sdot sdot sdot 119909

119896119899

119899 (47)

which implies for 119878119899= sum

119899

119894=1119883

119894 that

1198641198784

119899le 4 sdot sum

1198961+sdotsdotsdot+119896

119899=4

119864 (1198831198961

1sdot sdot sdot 119883

119896119899

119899)

le 4 sdot

119899

sum

119905=1

sum

0le119894119895119896le119899minus1

10038161003816100381610038161003816119864 (119883

119905119883

119905+119894119883

119905+119894+119895119883

119905+119894+119895+119896)10038161003816100381610038161003816

(48)

Let us now introduce the following notation due to Doukhanand Louhichi (see [14]) for a sequence of centered rvrsquos119883

119905119899

119899ge1

119862119903119902

= sup100381610038161003816100381610038161003816Cov (119883

1199051

sdot sdot sdot 119883119905119898

119883119905119898+119903sdot sdot sdot 119883

119905119902

)100381610038161003816100381610038161003816 (49)

where the supremum runs over all divisions of the groupcomposed of 119902 rvrsquos into two subgroups such that the distancebetween the highest index of the rvrsquos in the first group and thelowest index of the rvrsquos from the second group is equal to 119903119903 isin 1 2 For 119903 = 0 we shall define 119862

0119902= 1 Let us then

put

1198831199051

= 119868 [1198801199051

le 119909] minus 119909 1198831199052

= 119868 [1198801199052

le 119909] minus 119909 (50)

We will now estimate the summands in (48)If max119894 119895 119896 = 119894 then

10038161003816100381610038161003816119864 119883

119905sdot (119883

119905+119894119883

119905+119894+119895119883

119905+119894+119895+119896)10038161003816100381610038161003816

=100381610038161003816100381610038161003816Cov (119883

119905 119883

119905+119894119883

119905+119894+119895119883

119905+119894+119895+119896)100381610038161003816100381610038161003816le 119862

1198944

(51)

Similarly for max119894 119895 119896 = 119896 we have

10038161003816100381610038161003816119864 (119883

119905119883

119905+119894119883

119905+119894+119895) sdot 119883

119905+119894+119895+11989610038161003816100381610038161003816

=10038161003816100381610038161003816Cov (119883

119905119883

119905+119894119883

119905+119894+119895 119883

119905+119894+119895+119896)10038161003816100381610038161003816le 119862

1198964

(52)

When max119894 119895 119896 = 119895 then

10038161003816100381610038161003816119864 (119883

119905119883

119905+119894) sdot (119883

119905+119894+119895119883

119905+119894+119895+119896)10038161003816100381610038161003816

=10038161003816100381610038161003816Cov (119883

119905119883

119905+119894 119883

119905+119894+119895119883

119905+119894+119895+119896)

+119864 (119883119905119883

119905+119894) 119864 (119883

119905+119894+119895119883

119905+119894+119895+119896)10038161003816100381610038161003816

le 1198621198954

+ 1198621198942119862

1198962

(53)

We keep on estimating the fourth moment of 119878119899by intro-

ducing 119905 isin 0 119899mdashthe index for which |119864(119883119905119883

119905+119894119883

119905+119894+119895

119883119905+119894+119895+119896

)| attains its maximum

1198641198784

119899le 4119899 sum

0le119894119895119896le119899minus1

10038161003816100381610038161003816119864 (119883

119905119883

119905+119894119883

119905+119894+119895119883

119905+119894+119895+119896)10038161003816100381610038161003816

=4119899( sum

0le119894119896le119895

[1198621198942sdot 119862

1198962+ 119862

1198954]+ sum

0le119894119895le119896

1198621198964+ sum

0le119896119895le119894

1198621198944)

= 4119899( sum

0le119894119896le119895

1198621198942sdot 119862

1198962+ 3 sum

0le119896119895le119894

1198621198944)

(54)

6 ISRN Probability and Statistics

The terms obtained in (54) may further be bounded fromabove in the following way

sum

0le119894119896le119895

1198621198942sdot 119862

1198962=

119899minus1

sum

119895=0

119895

sum

119894119896=0

1198621198942119862

1198962

le 119899

119899minus1

sum

119894119896=0

1198621198942119862

1198962= 119899(

119899minus1

sum

119894=0

1198621198942)

2

sum

0le119896119895le119894

1198621198944

=

119899minus1

sum

119894=0

119894

sum

119895119896=0

1198621198944

le

119899minus1

sum

119894=0

(119894 + 1) (119894 + 1) 1198621198944

=

119899minus1

sum

119894=0

(119894 + 1)2119862

1198944

(55)

We thus get the inequality

1198641198784

119899le 4([119899

119899minus1

sum

119894=0

1198621198942]

2

+ 3119899

119899minus1

sum

119894=0

(119894 + 1)2119862

1198944) (56)

We shall now focus on estimating 1198621199032

and 1198621199034 Since 119880

1199051

and 1198801199052

are associated uniformly [0 1] distributed rvrsquos 1198831199051

and 1198831199052

mdashas monotone functions of these rvrsquosmdashare associat-ed as well In order to bound

1198621199032

= sup0le1199051lt1199052le119899 1199052minus1199051=119903

Cov (1198831199051

1198831199052

) (57)

let us notice that from Schwarz inequality

Cov (1198831199051

1198831199052

) = 119864 (119868 [1198801199051

le 119909] 119868 [1198801199052

le 119909]) minus 1199092

le radic1199092 minus 1199092le 119909

(58)

On the other hand invoking inequalities from [13 15]

Cov (1198831199051

1198831199052

)

le sup119909isin[01]

[119875 (1198801199051

le 1199091198801199052

le 119909) minus 119875 (1198801199051

le 119909)119875 (1198801199052

le 119909)]

le 119862 sdot Cov13

(1198801199051

1198801199052

) for associated rvrsquos4 sdot Cov (119880

1199051

1198801199052

) for MTP2rvrsquos

(59)

As a consequence

1198621199032

le min 119909 119862 sdot Cov13

(1198801199051

1198801199052

) for associated rvrsquosmin 119909 4 sdot Cov (119880

1199051

1198801199052

) for MTP2rvrsquos

(60)

where 1199052minus 119905

1= 119903 Still we need the upper bound for 119862

1199034 It

turns out that119862

1199034

= sup1003816100381610038161003816100381610038161003816100381610038161003816

Cov2

prod

119894=1

(119868 [119880119905119894

le 119909] minus 119909)

4

prod

119894=3

(119868 [119880119905119894

le 119909] minus 119909)

1003816100381610038161003816100381610038161003816100381610038161003816

(61)

In other words among all divisions of the group 119883119905119894

119894 isin

1 2 3 4 into two subgroups 1198621199034

attains the biggest valuein case we take two subgroups consisted of two rvrsquos Thesupremum runs over the set 119905

1 119905

2 119905

3 119905

4isin N 0 le 119905

1lt 119905

2lt

1199053lt 119905

4le 119899 and 119905

3minus 119905

2= 119903 Elementary calculation leads to the

following formula

1198621199034

= sup 10038161003816100381610038161003816119875 (119880119905119894

le 119909 119894 isin 1 2 3 4)

minus 119875 (119880119905119894

le 119909 119894 isin 1 2) 119875 (119880119905119894

le 119909 119894 isin 3 4)

minus 119909 [119875 (119880119905119894

le 119909 119894 isin 1 2 3)

minus 119875 (119880119905119894

le 119909 119894 isin 1 2) 119875 (1198801199053

le 119909)

+ 119875 (119880119905119894

le 119909 119894 isin 1 3 4)

minus 119875 (119880119905119894

le 119909 119894 isin 3 4) 119875 (1198801199051

le 119909)

+ 119875 (119880119905119894

le 119909 119894 isin 1 2 4)

minus 119875 (119880119905119894

le 119909 119894 isin 1 2) 119875 (1198801199054

le 119909)

+ 119875 (119880119905119894

le 119909 119894 isin 2 3 4)

minus119875 (119880119905119894

le 119909 119894 isin 3 4) 119875 (1198801199052

le 119909)]

+ 1199092[119875 (119880

1199051

le 1199091198801199053

le 119909)

minus 119875 (1198801199051

le 119909)119875 (1198801199053

le 119909)

+ 119875 (1198801199051

le 1199091198801199054

le 119909)

minus 119875 (1198801199051

le 119909)119875 (1198801199054

le 119909)

+ 119875 (1198801199052

le 1199091198801199053

le 119909)

minus 119875 (1198801199052

le 119909)119875 (1198801199053

le 119909)

+ 119875 (1198801199052

le 1199091198801199054

le 119909)

minus119875 (1198801199052

le 119909)119875 (1198801199052

le 119909)]10038161003816100381610038161003816

le10038161003816100381610038161198771

1003816100381610038161003816 +10038161003816100381610038161198772

1003816100381610038161003816 +10038161003816100381610038161198773

1003816100381610038161003816

(62)

where for the sake of simplicity 1198771 119877

2 and 119877

3are respec-

tively the free coefficient the expression with 119909 and theexpression with 119909

2 Using the Lebowitz inequality (see [15]for instance) and inequalities obtained in [13 15] we arrive at10038161003816100381610038161198771

1003816100381610038161003816 =10038161003816100381610038161003816119875 (119880

119905119894

le 119909 119894 isin 1 2 3 4)

minus119875 (119880119905119894

le 119909 119894 isin 1 2) 119875 (119880119905119894

le 119909 119894 isin 3 4)10038161003816100381610038161003816

le 11986711988011990511198801199053

+ 11986711988011990511198801199054

+ 11986711988011990521198801199053

+ 11986711988011990521198801199054

ISRN Probability and Statistics 7

le

119862(

2

sum

119894=1

4

sum

119895=3

Cov13(119880

119905119894

119880119905119895

)) for associated rvrsquos

4(

2

sum

119894=1

4

sum

119895=3

Cov (119880119905119894

119880119905119895

)) for MTP2rvrsquos

(63)

Analogously we get

10038161003816100381610038161198772

1003816100381610038161003816

le

2119909 sdot 119862(

2

sum

119894=1

4

sum

119895=3

Cov13(119880

119905119894

119880119905119895

)) for associated rvrsquos

2119909 sdot 4(

2

sum

119894=1

4

sum

119895=3

Cov (119880119905119894

119880119905119895

)) for MTP2rvrsquos

10038161003816100381610038161198773

1003816100381610038161003816

le

1199092sdot 119862(

2

sum

119894=1

4

sum

119895=3

Cov13(119880

119905119894

119880119905119895

) ) for associated rvrsquos

1199092sdot 4(

2

sum

119894=1

4

sum

119895=3

Cov (119880119905119894

119880119905119895

)) for MTP2rvrsquos

(64)

Eventually we have

1198621199034

le

119891 (119909) sdot119862(

2

sum

119894=1

4

sum

119895=3

Cov13(119880

119905119894

119880119905119895

)) for associated rvrsquos

119891 (119909) sdot4(

2

sum

119894=1

4

sum

119895=3

Cov (119880119905119894

119880119905119895

)) for MTP2rvrsquos

(65)

where 1199053minus 119905

2= 119903 and 119891(119909) = (1 + 2119909 + 119909

2) for 119909 isin [0 1]

Let us now introduce the following notation

120579119903=

sup119905isinN

Cov (119880119905 119880

119905+119903) for 119903 isin 1 2

1 for 119903 = 0

(66)

and assume it decays powerly at rate 119886 in the following way

120579119903= 119874(

1

(119903 + 1)119886) for 119886 gt 0 119903 isin 0 1 (67)

Let us get back to inequality (56) we can now carry on Atfirst for associated rvrsquos

1198641198784

119899le 4[119899

119899minus1

sum

119903=0

min 119909 119862 sdot Cov13(119880

1199051

1198801199052

)]

2

+ 4 sdot 3119899

119899minus1

sum

119903=0

(119903 + 1)2119891 (119909) sdot 119862(

2

sum

119894=1

4

sum

119895=3

Cov13(119880

119905119894

119880119905119895

))

le 119863 sdot ([119899

119899minus1

sum

119903=0

min 119909 (119903 + 1)minus1198863

]

2

+119899

119899minus1

sum

119903=0

(119903 + 1)2(119903 + 1)

minus1198863)

= 119863 sdot ([119899

119899

sum

119903=1

min 119909 119903minus1198863]

2

+ 119899

119899

sum

119903=1

1199032119903minus1198863

)

le 119863 sdot ([

[

119899 sum

119903lt119909minus3119886

119909 + 119899 sum

119903ge119909minus3119886

1

1199031198863]

]

2

+ 119899

119899

sum

119903=1

1199032minus1198863

)

le 1198631sdot (119899

2119909

2(119886minus3)119886+ 120585)

(68)

where119863 and1198631are constants and

120585 = 119899

119899

sum

119903=1

1199032minus1198863

=

119874(119899) 119886 gt 9

119874 (119899 ln 119899) 119886 = 9

119874 (1198994minus1198863

) 3 lt 119886 lt 9

(69)

It is worth mentioning that in the last inequality of (68) weused the estimate

int

infin

119909

1

119905119901119889119905 sim

1

119909119901minus1for 119901 gt 1 (70)

At the same time in the case of MTP2rvrsquos we get

1198641198784

119899le 119863

2sdot (119899

2119909

2(119886minus1)119886+ 120577) (71)

where1198632is constant and

120577 = 119899

119899

sum

119903=1

1199032minus119886

=

119874(119899) 119886 gt 3

119874 (119899 ln 119899) 119886 = 3

119874 (1198994minus119886

) 1 lt 119886 lt 3

(72)

Let now 120572119899(119909) = (1radic119899)sum

119899

119894=1(119868[119880

119905119894

le 119909] minus 119909) Then

120572119899(119909) minus 120572

119899(119910) =

1

radic119899

119899

sum

119894=1

(119868 [119909 lt 119880119905119894

le 119910] minus (119909 minus 119910))

for 119909 119910 isin [0 1]

(73)

8 ISRN Probability and Statistics

has the fourth moment estimatedmdashin the case of associatedrvrsquosmdashby

119864[120572119899(119909) minus 120572

119899(119910)]

4

= 119864[1

radic119899

119899

sum

119894=1

(119868 [119909 lt 119880119905119894

le 119910] minus (119909 minus 119910))]

4

=1

1198992119864[

119899

sum

119894=1

(119868 [119909 lt 119880119905119894

le 119910] minus (119909 minus 119910))]

4

le 1198631sdot (

1

119899211989921003816100381610038161003816119909 minus 119910

1003816100381610038161003816

2(119886minus3)119886

+120585

1198992)

= 1198631sdot (

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

2(119886minus3)119886

+120585

1198992)

=

119874(119899minus1) 119886 gt 9

119874(ln 119899119899

) 119886 = 9

119874 (1198992minus1198863

) 3 lt 119886 lt 9

(74)

and in the case of MTP2rvrsquos by

119864[120572119899(119909) minus 120572

119899(119910)]

4

le 1198632sdot (

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

2(119886minus1)119886

+120577

1198992)

=

119874(119899minus1) 119886 gt 3

119874(ln 119899119899

) 119886 = 3

119874 (1198992minus119886

) 1 lt 119886 lt 3

(75)

In light of the Shao and Yursquos criterion our process is tight forassociated rvrsquos when 119886 gt 6 and for MTP

2rvrsquos when 119886 gt 2

Let us sum up this result in the following theorem

Theorem 3 Let 120572119899(119909) = (1radic119899)sum

119899

119894=1(119868[119880

119894le 119909] minus 119909) be the

empirical process built on an associated sequence of uniformly[0 1] distributed rvrsquos 119880

119894119894ge1

Let also

120579119903= sup

119905isinNCov (119880

119905 119880

119905+119903) = 119874 ((119903 + 1)

minus119886)

where 119903 isin 0 1 119886 gt 0

(76)

Then 120572119899(119909)

119899ge1is tight for 119886 gt 6 If the rvrsquos 119880

119894119894ge1

are MTP2

then the process is tight for 119886 gt 2

Yu assumed stationarity of 119880119894119894ge1

andsum

infin

119899=111989965+]Cov(119880

0 119880

119899) lt infin for a positive constant ]

thus the rate of decay 119886 gt 7 5 Our result weakens consid-erably these assumptions especially in the case of MTP

2rvrsquos

Louhichi in [16] proposed a different tightness criterioninvolving the so-called bracketing numbers She managedto enhance Yursquos resultmdasheven more than Shao and Yu in[8]mdashsince she proved that it suffices to take 119886 gt 4 to gettightness of the empirical process based on the associated

rvrsquos Nevertheless she kept the assumption of stationarityvalid

In the final analysis our resultrsquos advantage is the absenceof the stationarity assumption and the rate of decay for120579119903remains (up to the authorrsquos knowledge) unimproved for

MTP2rvrsquos

Unfortunately with a view to obtainingweak convergenceof the process in question that is also convergence offinite-dimensional distributions we do not know how tomanage without the assumption of stationarityTherefore weconclude with the following corollary

Corollary 4 Let 120572119899(119909) = (1radic119899)sum

119899

119894=1(119868[119880

119894le 119909] minus 119909) be the

empirical process built on a stationary associated sequence ofuniformly [0 1] distributed rvrsquos 119880

119894119894ge1

Let also

120579119903= Cov (119880

1 119880

1+119903) = 119874 ((119903 + 1)

minus119886)

where 119903 isin 0 1 119886 gt 0

(77)

Then if 119886 gt 6

120572119899(sdot) 997888rarr 119861 (sdot) weakly in 119863 [0 1] (78)

where 119861(sdot) is the zero mean Gaussian process on [0 1] withcovariance structure defined by

1205902(119909 119910) = 119909 and 119910 minus 119909119910

+

infin

sum

119895=1

[ Cov (119868 [1198801le 119909] 119868 [119880

119895+1le 119910])

+ Cov (119868 [1198801le 119910] 119868 [119880

119895+1le 119909])]

(79)

If the rvrsquos 119880119894119894ge1

are MTP2 then it suffices to be 119886 gt 2 in

order to claim the above convergence

Proof It remains to establish convergence of finite-dimen-sional distributions repeating the procedure from [7]

4 Tightness of the Kernel-TypeEmpirical Process

In this section we shall weaken assumption imposed on thecovariance structure of rvrsquos 119883

119895119895ge1

by Cai and Roussas in [1]for the kernel estimator of the df

119865119899(119909) =

1

119899

119899

sum

119895=1

119870(

119909 minus 119883119895

ℎ119899

) (80)

They deal with a stationary sequence of negatively associatedrvrsquos (cf [17]) and need the same condition as Yu [7] that is

1003816100381610038161003816Cov (1198831 119883

1+119899)1003816100381610038161003816 = 119874(

1

11989975+]) (81)

to get tightness of the smooth empirical process (see condi-tion (A4) in [1])

ISRN Probability and Statistics 9

It turns out that it suffices to have

1003816100381610038161003816Cov (1198801 119880

1+119899)1003816100381610038161003816 = 119874(

1

1198993119901(119901minus1)) (82)

where 119901 gt 4 is a positive constant taken from the tightnesscriterion (8) It is easy to see that asymptotically we get therate 3

On the way to prove it we will also take use of aRosenthal-type inequality due to Shao andYu (seeTheorem 2in [8]) we shall recall in the following lemma

Lemma 5 Let 119901 gt 2 and 119891 be a real valued function boundedby 1 with bounded first derivative Suppose that 119883

119899119899ge1

is asequence of stationary and associated rvrsquos such that for 119899 isin N

Cov (1198831 119883

119899) = 119874 (119899

minus119887) for some 119887 gt 119901 minus 1 (83)

Then for any 120583 gt 0 there exists some positive constant 119896120583

independent of the function 119891 for which

1198641003816100381610038161003816119878119899

(119891) minus 119864119878119899(119891)

1003816100381610038161003816

119901

le 119896120583(119899

1+12058310038171003817100381710038171003817119891

101584010038171003817100381710038171003817

2

+1198991199012 [

[

119899

sum

119895=1

10038161003816100381610038161003816Cov (119891 (119883

1)119891 (119883

119895))10038161003816100381610038161003816]

]

)

(84)

As we can see the lemma assumes association but itworks for negatively associated rvrsquos as well since in the proofit reaches back the result of Newman (see Proposition 15 in[18]) where both types of association are allowed

Let us recall that 120572119899(119909) = radic119899(119865

119899(119909) minus 119864119865

119899(119909)) where

119865119899(119909) = (1119899)sum

119899

119895=1119870((119909 minus 119883

119895)ℎ

119899) It is easy to see that

1198641003816100381610038161003816120572119899

(119909) minus 120572119899(119910)

1003816100381610038161003816

119901

= 119899minus1199012

11986410038161003816100381610038161003816119878119899(119891 (119883

119895)) minus 119864119878

119899(119891 (119883

119895))10038161003816100381610038161003816

119901

(85)

where

119891 (119883119895) = 119870(

119909 minus 119883119895

ℎ119899

) minus 119870(

119910 minus 119883119895

ℎ119899

)

119878119899(119891) =

119899

sum

119895=1

119891 (119883119895)

(86)

With an intent to use Lemma 5 we need 119891 to be boundedfrom above by 1 (which in our case is obvious) and to havebounded 119891

1015840 thus we assume

sup119905isinR

10038161003816100381610038161003816119870

1015840(119905)

10038161003816100381610038161003816

1

ℎ119899

le119862

119870

ℎ119899

lt infin (87)

Now applying inequality (84) we have

1198641003816100381610038161003816120572119899

(119909) minus 120572119899(119910)

1003816100381610038161003816

119901

le119896120583119899minus1199012

1198991+120583

1198622

119870

ℎ2

119899

+1198991199012[

[

119899

sum

119895=1

10038161003816100381610038161003816Cov (119891 (119883

1)119891 (119883

119895))10038161003816100381610038161003816]

]

1199012

(88)

Newman showed in [18] that

Cov (1198911(119883) 119891

2(119884))

= ∬R2119891

1015840

1(119909) 119891

1015840

2(119910)

times [119875 (119883 le 119909 119884 le 119910) minus 119875 (119883 le 119909) 119875 (119884 le 119910)] 119889119909 119889119910

(89)

if 1198911and 119891

2are real valued functions on R having square

integrable derivatives 1198911015840

1and 119891

1015840

2 respectively and provided

that 1198911(119883) 119891

2(119884) have finite secondmoments In light of that

equation and linearity of covariance we get10038161003816100381610038161003816Cov (119891 (119883

1) 119891 (119883

119895))10038161003816100381610038161003816

=

1003816100381610038161003816100381610038161003816∬

R2[119875 (119883

1lt 119909 minus ℎ

119899119904 119883

119895lt 119909 minus ℎ

119899119905)

minus119875 (1198831lt 119909 minus ℎ

119899119904 119883

119895lt 119910 minus ℎ

119899119905)]

minus [119875 (1198831lt 119909 minus ℎ

119899119904) 119875 (119883

119895lt 119909 minus ℎ

119899119905)

minus119875 (1198831lt 119909 minus ℎ

119899119904) 119875 (119883

119895lt 119910 minus ℎ

119899119905)]

minus [119875 (1198831lt 119910 minus ℎ

119899119904 119883

119895lt 119909 minus ℎ

119899119905)

minus119875 (1198831lt 119910 minus ℎ

119899119904 119883

119895lt 119910 minus ℎ

119899119905)]

minus [119875 (1198831lt 119910 minus ℎ

119899119904) 119875 (119883

119895lt 119910 minus ℎ

119899119905)

minus 119875 (1198831lt 119910 minus ℎ

119899119904)

times 119875 (119883119895lt 119909 minus ℎ

119899119905)] 119896 (119904) 119896 (119905) 119889119904 119889119905

10038161003816100381610038161003816100381610038161003816

(90)

Without the loss of generality we may and do assume that119909 gt 119910 thus10038161003816100381610038161003816Cov (119891 (119883

1) 119891 (119883

119895))10038161003816100381610038161003816

=

1003816100381610038161003816100381610038161003816∬

R2119875 (119910 minus ℎ

119899119904 le 119883

1lt 119909 minus ℎ

119899119904

119910 minus ℎ119899119905 le 119883

119895lt 119909 minus ℎ

119899119905)

minus 119875 (119910 minus ℎ119899119904 le 119883

1lt 119909 minus ℎ

119899119904)

times 119875 (119910 minus ℎ119899119905 le 119883

119895lt 119909 minus ℎ

119899119905) 119896 (119904) 119896 (119905) 119889119904 119889119905

1003816100381610038161003816100381610038161003816

(91)

and by triangle inequality we get10038161003816100381610038161003816Cov (119891 (119883

1) 119891 (119883

119895))10038161003816100381610038161003816

le ∬R2[int

119909minusℎ119899119904

119910minusℎ119899119904

int

119909minusℎ119899119905

119910minusℎ119899119905

1198911198831119883119895(119906 V) 119889119906 119889V

+int

119909minusℎ119899119904

119910minusℎ119899119904

1198911198831(119906) 119889119906int

119909minusℎ119899119905

119910minusℎ119899119905

119891119883119895(V) 119889V] 119896 (119904) 119896 (119905) 119889119904 119889119905

(92)

10 ISRN Probability and Statistics

1198911198831119883119895

(119906 V) 1198911198831

(119906) and 119891119883119895

(V) are the joint pdf of [1198831 119883

119895]

and marginal pdfrsquos of rvrsquos1198831and119883

119895 respectively We need

to further assume that 1198621stands for a common upper bound

of 1198911198831

and 1198911198831119883119895

for all 119895 isin N that is

100381710038171003817100381710038171198911198831

10038171003817100381710038171003817le 119862

1

1003817100381710038171003817100381710038171198911198831119883119895

100381710038171003817100381710038171003817le 119862

1forall119895 ge 1 (93)

Then we finally obtain

10038161003816100381610038161003816Cov (119891 (119883

1) 119891 (119883

119895))10038161003816100381610038161003816le 2119862

1

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

2

where = max 1198621 119862

2

1

(94)

On the other hand proceeding like Cai and Roussas in [1] wecan shortly get

10038161003816100381610038161003816Cov (119891 (119883

1) 119891 (119883

119895))10038161003816100381610038161003816

le

100381610038161003816100381610038161003816100381610038161003816

Cov(119870(119909 minus 119883

1

ℎ119899

) 119870(

119909 minus 119883119895

ℎ119899

))

100381610038161003816100381610038161003816100381610038161003816

+

100381610038161003816100381610038161003816100381610038161003816

Cov(119870(119909 minus 119883

1

ℎ119899

) 119870(

119910 minus 119883119895

ℎ119899

))

100381610038161003816100381610038161003816100381610038161003816

+

100381610038161003816100381610038161003816100381610038161003816

Cov(119870(119910 minus 119883

1

ℎ119899

) 119870(

119909 minus 119883119895

ℎ119899

))

100381610038161003816100381610038161003816100381610038161003816

+

100381610038161003816100381610038161003816100381610038161003816

Cov(119870(119910 minus 119883

1

ℎ119899

) 119870(

119910 minus 119883119895

ℎ119899

))

100381610038161003816100381610038161003816100381610038161003816

le 41198622

10038161003816100381610038161003816Cov (119883

1 119883

119895)10038161003816100381610038161003816

13

(95)

where 1198622is a constant relevant to the covariance inequality

for negatively associated rvrsquos (see [13])We now arrive at the following inequality

10038161003816100381610038161003816Cov (119891 (119883

1) 119891 (119883

119895))10038161003816100381610038161003816

le max 1198621 119862

2

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

2

10038161003816100381610038161003816Cov(119883

1 119883

119895)10038161003816100381610038161003816

13

(96)

Assuming |Cov(119891(1198831) 119891(119883

119895))| = 119874(119895

minus119903) and proceeding

similarly to (68) we can get

[

[

119899

sum

119895=1

10038161003816100381610038161003816Cov (119891 (119883

1) 119891 (119883

119895))10038161003816100381610038161003816]

]

1199012

le max 1198621 119862

2 [

[

sum

119895lt|119909minus119910|minus2(1199033)

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

2

+ sum

119895ge|119909minus119910|minus2(1199033)

1

1198951199033]

]

1199012

le 2max 1198621 119862

21003816100381610038161003816119909 minus 119910

1003816100381610038161003816

((119903minus3)119903)119901

(97)

To sum up we obtain the following inequality

1198641003816100381610038161003816120572119899

(119909) minus 120572119899(119910)

1003816100381610038161003816

119901

le 119896120583119899

1+120583minus1199012119862

2

119870

ℎ2

119899

+ 2max 1198621 119862

21003816100381610038161003816119909 minus 119910

1003816100381610038161003816

((119903minus3)119903)119901

le 1198621198991+120583minus1199012 1

ℎ2

119899

+1003816100381610038161003816119909 minus 119910

1003816100381610038161003816

((119903minus3)119903)119901

(98)

where 119862 = 119896120583max1198622

119896 2max119862

1 119862

2 The formula of the

tightness criterion (8) implies that ((119903 minus 3)119903)119901 gt 1 so

119903 gt3119901

119901 minus 1 where 119901 gt 2 (99)

Let us recall the assumptions imposed on the bandwidhtsℎ

119899119899ge1

by Cai and Roussas in [1]B1 lim

119899rarrinfinℎ119899= 0 and ℎ

119899gt 0 for all 119899 isin N

+

B2 lim119899rarrinfin

119899ℎ119899= infin thus ℎ

119899= 119874(1119899

1minus120573) 120573 gt 0

B3 lim119899rarrinfin

119899ℎ4

119899= 0 hence ℎ

119899= 119874(1119899

14+120575) 120575 gt 0

In light of the above

1198991+120583minus1199012 1

ℎ2

119899

= 11989932+120583+2120575minus1199012

(100)

where 120583 gt 0 119901 gt 2 and 0 lt 120575 lt 34 Confrontation with thetightness criterion (8) forces

3

2+ 120583 + 2120575 minus

119901

2lt minus

1

2 (101)

which implies 119901 gt 4 Let us conclude with the followingtheorem

Theorem6 Let 120572119899(119909) = radic119899(119865

119899(119909)minus119864119865

119899(119909)) be the empirical

process built on the kernel estimator of the df119865 for a stationarysequence of negatively associated rvrsquos Assume that conditionsA1 A2 (87) B1 B2 B3 are satisfied Then provided that thetightness criterion (8) holds with 119901 gt 4 it suffices to demand

1003816100381610038161003816Cov (1198801 119880

1+119899)1003816100381610038161003816 = 119874(

1

1198993119901(119901minus1)) (102)

in order to obtain120572

119899(sdot) 997888rarr 119861 (sdot) weakly in 119863 [0 1] (103)

where 119861(sdot) is the zero mean Gaussian process with covariancestructure defined by

1205902(119909 119910) = 119865 (119909 and 119910) minus 119865 (119909) 119865 (119910)

+

infin

sum

119895=1

[ Cov (119868 [1198831le 119909] 119868 [119883

119895+1le 119910])

+ Cov (119868 [1198831le 119910] 119868 [119883

119895+1le 119909])]

(104)

and 119909 119910 isin [0 1]

Proof The remaining covergence of finite-dimensional dis-tributions of 120572

119899(119909) is established in [1]

ISRN Probability and Statistics 11

5 Weak Convergence of theRecursive Kernel-Type Empirical Processunder IID Assumption

The aim of this section is to show weak convergence of theempirical process

120572119899(119909) =

1

radic119899

119899

sum

119895=1

[119870(

119909 minus 119883119895

ℎ119895

) minus 119864119870(

119909 minus 119883119895

ℎ119895

)] (105)

built on iid rvrsquos 119883119895119895ge1

Wewill prove that120572

119899(sdot) convergesweakly to the Brownian

bridge 119861(sdot) with the following covariance structure

1205902(119909 119910) = 119865 (119909 and 119910) minus 119865 (119909) 119865 (119910) for 119909 119910 isin [0 1]

(106)

It turns out that the tightness criterion (8) suffices to reachthe goal Convergence of finite-dimensional distributions of120572

119899(sdot) holds and we shall show itProceeding like Cai and Roussas in [1] we need to show

that for any 119886 119887 isin R

119886120572119899(119909) + 119887120572

119899(119910) 997888rarr 119886119861 (119909) + 119887119861 (119910) in distribution

(107)

Let us introduce the notation

119884119894(119909) = 119870(

119909 minus 119883119894

ℎ119894

) minus 119864119870(119909 minus 119883

119894

ℎ119894

) (108)

and look into the covariance structure of 120572119899(sdot)

Cov (120572119899(119909) 120572

119899(119910))

= Cov( 1

radic119899

119899

sum

119894=1

119884119894(119909)

1

radic119899

119899

sum

119894=1

119884119894(119910))

=1

119899

119899

sum

119894=1

Cov (119884119894(119909) 119884

119894(119910))

=1

119899

119899

sum

119894=1

Cov(119870(119909 minus 119883

119894

ℎ119894

) 119870(119910 minus 119883

119894

ℎ119894

))

(109)

where the second equality follows from assumed indepen-dence of rvrsquos 119883

119894119894ge1

Firstly we observe that each summand converges to119865(119909and

119910) minus 119865(119909)119865(119910) since

Cov(119870(119909 minus 119883

119894

ℎ119894

) 119870(119910 minus 119883

119894

ℎ119894

))

= 119864[119870(119909 minus 119883

119894

ℎ119894

)119870(119910 minus 119883

119894

ℎ119894

)]

minus 119864119870(119909 minus 119883

119894

ℎ119894

)119864119870(119910 minus 119883

119894

ℎ119894

)

(110)

where

119864[119870(119909 minus 119883

119894

ℎ119894

)119870(119910 minus 119883

119894

ℎ119894

)]

= int

119909and119910

minusinfin

119870(119909 minus 119905

ℎ119894

)119870(119910 minus 119905

ℎ119894

)119889119865 (119905)

+ int

119909or119910

119909and119910

119870(119909 minus 119905

ℎ119894

)119870(119910 minus 119905

ℎ119894

)119889119865 (119905)

+ int

infin

119909or119910

119870(119909 minus 119905

ℎ119894

)119870(119910 minus 119905

ℎ119894

)119889119865 (119905)

(111)

and 119865 denotes the common df of the rvrsquos 119883119894119894ge1

In [6] itwas shown that

119864119870(119909 minus 119883

119894

ℎ119894

) 997888rarr 119865 (119909) (112)

and recalling that the kernel function 119870 is a df as well wearrive at the conclusion

Secondly applying Toeplitz lemma we get

Cov (120572119899(119909) 120572

119899(119910)) 997888rarr 119865 (119909 and 119910) minus 119865 (119909) 119865 (119910)

= 1205902(119909 119910) 119899 997888rarr infin

(113)

Thus

Var (119886120572119899(119909) + 119887120572

119899(119910))

= 1198862 Var (120572

119899(119909)) + 2119886119887Cov (120572

119899(119909) 120572

119899(119910))

+ 1198872 Var (120572

119899(119910))

997888rarr 1198862120590

2(119909 119909) + 2119886119887120590

2(119909 119910) + 119887

2120590

2(119910 119910) 119899 997888rarr infin

(114)

We are now on the way to prove that 119886120572119899(119909) + 119887120572

119899(119910) con-

verges in distribution to 119886119861(119909) + 119887119861(119910) sim N(0 1198862120590

2(119909 119909) +

21198861198871205902(119909 119910) + 119887

2120590

2(119910 119910)) which in other words means that

1

radic119899

infin

sum

119894=1

119886119884119894(119909) + 119887119884

119894(119910)

radicVar (119886120572119899(119909) + 119887120572

119899(119910))

997888rarr N (0 1) (115)

In order to obtain (115) we will use Lyapunov condition

lim119899rarrinfin

1

1198632+120575

119899

119899

sum

119894=1

1198641003816100381610038161003816119886119884119894

(119909) + 119887119884119894(119910)

1003816100381610038161003816

2+120575

= 0 (116)

for 120575 = 1 where 1198632

119899= Var(sum119899

119894=1[119886119884

119894(119909) + 119887119884

119894(119910)]) Since

|119870(sdot) minus 119864119870(sdot)| le 2

1198641003816100381610038161003816119886119884119894

(119909) + 119887119884i (119910)1003816100381610038161003816

3

le 8 (1198863+ 119887

3) (117)

Moreover 1198632

119899= 119899Var(119886120572

119899(119909) + 119887120572

119899(119910)) together with (114)

yields

1198633

119899= 119874 (119899

32) (118)

12 ISRN Probability and Statistics

Combining (117) with (118) we obtain

1

1198633

119899

119899

sum

119894=1

1198641003816100381610038161003816119886119884119894

(119909) + 119887119884119894(119910)

1003816100381610038161003816

3

= 119874 (119899minus12

) (119)

which shows that Lyapunov condition holds and completesthe proof of convergence of finite-dimensional distributionsof 120572

119899(119909)We summarize the result of that section in the following

theorem

Theorem 7 Let 120572119899(119909) be the recursive kernel-type empirical

process defined by (105) built on iid rvrsquos 119883119895119895ge1

withmarginal df 119865 If 120572

119899(sdot) satisfies the tightness criterion (8) then

120572119899(sdot) 997888rarr 119861 (sdot) weakly in 119863 [0 1] (120)

where 119861 is the Brownian bridge

Acknowledgment

The author would like to thank the referees for the carefulreading of the paper and helpful remarks

References

[1] Z Cai and G G Roussas ldquoWeak convergence for smoothestimator of a distribution function under negative associationrdquoStochastic Analysis and Applications vol 17 no 2 pp 145ndash1681999

[2] Z Cai andG G Roussas ldquoEfficient Estimation of a DistributionFunction under Qudrant Dependencerdquo Scandinavian Journal ofStatistics vol 25 no 1 pp 211ndash224 1998

[3] ZW Cai and G G Roussas ldquoUniform strong estimation under120572-mixing with ratesrdquo Statistics amp Probability Letters vol 15 no1 pp 47ndash55 1992

[4] G G Roussas ldquoKernel estimates under association stronguniform consistencyrdquo Statistics amp Probability Letters vol 12 no5 pp 393ndash403 1991

[5] Y F Li and S C Yang ldquoUniformly asymptotic normalityof the smooth estimation of the distribution function underassociated samplesrdquo Acta Mathematicae Applicatae Sinica vol28 no 4 pp 639ndash651 2005

[6] Y Li C Wei and S Yang ldquoThe recursive kernel distributionfunction estimator based onnegatively and positively associatedsequencesrdquo Communications in Statistics vol 39 no 20 pp3585ndash3595 2010

[7] H Yu ldquoA Glivenko-Cantelli lemma and weak convergence forempirical processes of associated sequencesrdquo ProbabilityTheoryand Related Fields vol 95 no 3 pp 357ndash370 1993

[8] Q M Shao and H Yu ldquoWeak convergence for weightedempirical processes of dependent sequencesrdquo The Annals ofProbability vol 24 no 4 pp 2098ndash2127 1996

[9] P Billingsley Convergence of Probability Measures Wiley Seriesin Probability and Statistics Probability and Statistics JohnWiley amp Sons New York NY USA 1999

[10] J D Esary F Proschan and D W Walkup ldquoAssociation ofrandom variables with applicationsrdquo Annals of MathematicalStatistics vol 38 pp 1466ndash1474 1967

[11] A Bulinski and A Shashkin Limit Theorems for AssociatedRandom Fields and Related Systems vol 10 of Advanced Serieson Statistical Science and Applied Probability World Scientific2007

[12] S Karlin and Y Rinott ldquoClasses of orderings of measures andrelated correlation inequalities I Multivariate totally positivedistributionsrdquo Journal of Multivariate Analysis vol 10 no 4 pp467ndash498 1980

[13] P Matuła and M Ziemba ldquoGeneralized covariance inequali-tiesrdquo Central European Journal of Mathematics vol 9 no 2 pp281ndash293 2011

[14] P Doukhan and S Louhichi ldquoA new weak dependence condi-tion and applications to moment inequalitiesrdquo Stochastic Proc-esses and their Applications vol 84 no 2 pp 313ndash342 1999

[15] P Matuła ldquoA note on some inequalities for certain classes ofpositively dependent random variablesrdquo Probability and Math-ematical Statistics vol 24 no 1 pp 17ndash26 2004

[16] S Louhichi ldquoWeak convergence for empirical processes of asso-ciated sequencesrdquo Annales de lrsquoInstitut Henri Poincare vol 36no 5 pp 547ndash567 2000

[17] K Joag-Dev and F Proschan ldquoNegative association of randomvariables with applicationsrdquoThe Annals of Statistics vol 11 no1 pp 286ndash295 1983

[18] C M Newman ldquoAsymptotic independence and limit theoremsfor positively and negatively dependent random variablesrdquo inInequalities in Statistics and Probability Y L Tong Ed vol 5pp 127ndash140 Institute ofMathematical StatisticsHayward CalifUSA 1984

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

ISRN Probability and Statistics 3

where

119865119899(119909) =

1

119899

119899

sum

119895=1

119870(

119909 minus 119883119895

ℎ119899119895

) (16)

is the generalized kernel estimator of the df119865 ApplyingChe-byshevrsquos inequality we shall find 119886 satisfying

119886 geradic119899

120578sdot 119864

10038161003816100381610038161003816119865119899(119909) minus 119864119865

119899(119909)

10038161003816100381610038161003816forall119899 isin N (17)

Such 119886 exists if 119864|120572119899(119909)| lt infin for all 119899 ge 1 Implementing

119910 = 0 and 1199013= 0 in the assumption (8) we get for 119901 gt 2

1199011 119901

2gt 1

1198641003816100381610038161003816120572119899

(119909)1003816100381610038161003816

119901

le 119862 sdot(|119909|1199011+119899

minus11990122|119909|

1199013) le 119862 sdot(1+

1

11989911990122) le 2119862

(18)

Now applying Holderrsquos inequality we arrive at

1198641003816100381610038161003816120572119899

(119909)1003816100381610038161003816 le

4radic119864

1003816100381610038161003816120572119899(119909)

1003816100381610038161003816

4

(19)

which in light of the inequality (18) for 119901 = 4 is boundedfrom above by 4radic2119862 lt infin Therefore condition (9) holds

We shall now proceed to checking condition (10) Let usrecall that the modulus of continuity of the function 119909 isin

119862[0 1] 120575 gt 0 is given by the formula

1199081(119909 120575) = sup

0le119910le1minus120575

sup119910le119909le119910+120575

1003816100381610038161003816120572119899(119909) minus 120572

119899(119910)

1003816100381610038161003816 (20)

As it is shown in [9] (page 123) there is a relation between119908

1(sdot sdot) and 1199081015840

1(sdot sdot) in the spaces 119862[0 1] and119863[0 1] relatively

Namely for 119909 isin 119863[0 1] and 120575 lt 12

1199081(119909 2120575) ge 119908

1015840

1(119909 120575) (21)

Thus

119875 (1199081015840

1(120572

119899 120575) ge 120598) le 119875 (119908

1(120572

119899 2120575) ge 120598) (22)

Therefore condition (10) holds if we show

forall120598gt0

forall120578gt0

exist0le120575lt12

exist1198990

forall119899ge1198990

119875 (1199081(120572

119899 2120575) ge 120598) le 120578 (23)

With a view to obtaining the desired inequality we shall pro-ceed patiently in five steps

Step 1 We need a moment inequality for the rv 120572119899(119909)minus120572

119899(119910)

involving the distance between the points 119909 and 119910 Let usthen assume that for the constants 119862 119901 119901

1 119901

2 119901

3given in

Theorem 1 inequality (8) holds Fix 120598 gt 0 120578 gt 0 and definethe quantity 119903

119899= 120598radic119899 119899 isin N Next we fix 119909 119910 isin [0 1] and

take 119899 large enough so that 119903119899le |119909minus119910|Then 119899minus12

le |119909minus119910|120598minus1

and we have

1198641003816100381610038161003816120572119899

(119909) minus 120572119899(119910)

1003816100381610038161003816

119901

le 119862 (1 + 120598minus1199012)1003816100381610038161003816119909 minus 119910

1003816100381610038161003816

min11990111199012+1199013

(24)

Step 2 Let us now fix 119899 isin N 120575 isin (0 1) and consider the fol-lowing rvrsquos

120594119894= 120572

119899(119910 + 119894119903

119899) minus 120572

119899(119910 + (119894 minus 1) 119903

119899) (25)

for 119894 isin 1 2 119898119899 where119898

119899= 119898(119899 120575) is such that 119903

119899119898

119899le 120575

It is easy to see that for 119878119894= sum

119894

119896=1120594

119896

max1le119894le119898

119899

1003816100381610038161003816119878119894

1003816100381610038161003816 = max1le119894le119898

119899

1003816100381610038161003816120572119899(119910 + 119894119903

119899) minus 120572

119899(119910)

1003816100381610038161003816 (26)

Let us notice that for rvrsquos 120594119894119894ge1

the conditions of Theo-rem 102 in [9] are satisfied with 4120573 = 119901 119906

119897= 119903

119899forall

119894lt119897le119895and

2120572 = min1199011 119901

2+ 119901

3 Therefore we are equipped with the

following maximal inequality

119875( max1le119894le119898

119899

1003816100381610038161003816120572119899(119910 + 119894119903

119899) minus 120572

119899(119910)

1003816100381610038161003816 ge 120582)

le

119862119901min119901

11199012+1199013

120582119901(119898

119899119903119899)min119901

11199012+1199013

(27)

for all 120582 gt 0

Step 3 Let 119872 = sup119909isin[01]

119891(119909) where 119891 is the pdf of rvrsquos119883

119895119895ge1

For fixed 120598 120578 gt 0 let us take 120575 gt 0 such that

119862119901min119901

11199012+1199013

(119872120598)119901

sdot (2120575)min119901

11199012+1199013lt 120578 (28)

and define119898119899= lfloor120575119903

119899rfloor For sufficiently large 119899 isin N we have

119898119899ge 1 and then we get

119903119899119898

119899le 120575 lt (119898

119899+ 1) 119903

119899le 2119898

119899119903119899le 120575 (29)

Step 4 Our goal is to obtain an inequality which enables usto bound the supremum of the increment of the function 120572

119899

via the maximum of the increments of that function on somesubintervals To bemore precise wewill find the upper boundfor

sup119910le119909le119910+119898

119899119902

1003816100381610038161003816120572119899(119909) minus 120572

119899(119910)

1003816100381610038161003816 (30)

in terms of

max1le119894le119898

119899

1003816100381610038161003816120572119899(119910 + 119894119902) minus 120572

119899(119910)

1003816100381610038161003816 (31)

where 119909 119910 isin [0 1] 119910 le 119909 le 119910 + 119902 119902 ge 120598119899 and119898119899is defined

as in Step 3Let us recall that 120572

119899(119909) = radic119899(119865

119899(119909) minus 119864119865

119899(119909)) where

119865119899(119909) = (1119899)sum

119899

119895=1119870((119909 minus 119883

119895)ℎ

119899119895) From the triangle ine-

quality we can see that

1003816100381610038161003816120572119899(119909) minus 120572

119899(119910)

1003816100381610038161003816

leradic119899

119899

10038161003816100381610038161003816100381610038161003816100381610038161003816

119899

sum

119895=1

[119870(

119909 minus 119883119895

ℎ119899119895

) minus 119870(

119910 minus 119883119895

ℎ119899119895

)]

10038161003816100381610038161003816100381610038161003816100381610038161003816

+ radic119899 (10038161003816100381610038161003816119864119865

119899(119909) minus 119865 (119909)

10038161003816100381610038161003816+10038161003816100381610038161003816119864119865

119899(119910) minus 119865 (119910)

10038161003816100381610038161003816

+1003816100381610038161003816119865 (119909) minus 119865 (119910)

1003816100381610038161003816 )

(32)

4 ISRN Probability and Statistics

Since 119870 is a nondecreasing function and applying triangleinequality again we get

1003816100381610038161003816120572119899(119909) minus 120572

119899(119910)

1003816100381610038161003816

le1003816100381610038161003816120572119899

(119910 + 119902) minus 120572119899(119910)

1003816100381610038161003816 +radic119899

10038161003816100381610038161003816119864119865

119899(119910 + 119902) minus 119864119865

119899(119910)

10038161003816100381610038161003816

+ radic119899 (10038161003816100381610038161003816119864119865

119899(119909) minus 119865 (119909)

10038161003816100381610038161003816+10038161003816100381610038161003816119864119865

119899(119910) minus 119865 (119910)

10038161003816100381610038161003816

+1003816100381610038161003816119865 (119909) minus 119865 (119910)

1003816100381610038161003816 )

(33)

Cai and Roussas in [1] showed that under assumptions madeon the df 119865 and the kernel function119870 we have

10038161003816100381610038161003816119864119865

119899(119909) minus 119865 (119909)

10038161003816100381610038161003816= 119874 (ℎ

2

119899) (34)

Similarly by Taylor expansion it is easy to see that

1003816100381610038161003816119865 (119909) minus 119865 (119910)1003816100381610038161003816 le

10038171003817100381710038171198911003817100381710038171003817 sdot

1003816100381610038161003816119909 minus 1199101003816100381610038161003816 (35)

Thus

1003816100381610038161003816120572119899(119909) minus 120572

119899(119910)

1003816100381610038161003816 le1003816100381610038161003816120572119899

(119910 + 119902) minus 120572119899(119910)

1003816100381610038161003816

+ 119862radic119899ℎ4

119899+119872radic119899119902

(36)

where 119872 = sup119909isin[01]

119891(119909) and 119862 = 1198621198911015840119870

is a positive con-stant dependent on the functions 119865 and119870

Let us recall that 119910 le 119909 le 119910+ 119902119898119899= lfloor120575119903

119899rfloor and 119902 ge 120598119899

Taking 119902 = (120598radic119899)(= 119903119899) we notice that 119898

119899119902 = lfloor120575119903

119899rfloor119903

119899le 120575

thus when 119899 rarr infin the interval [119910 119910 + 119898119899119902] coincides with

[119910 119910 + 120575] Approaching the aim of Step 4 let us observe that

sup119910le119909le119910+119898

119899119902

1003816100381610038161003816120572119899(119909) minus 120572

119899(119910)

1003816100381610038161003816 = max1le119894le119898

119899

1003816100381610038161003816120572119899(119910 + 119894119902) minus 120572

119899(119910)

1003816100381610038161003816

+1003816100381610038161003816120572119899

(1199090) minus 120572

119899(119910 + 119894max119902)

1003816100381610038161003816

(37)

where 1199090isin [119910+119894

0119902 119910+(119894

0+1)119902] is a point at which the above

supremum is attained and

119894max =

1198940+ 1 if 1003816100381610038161003816120572119899

(119910 + [1198940+ 1] 119902) minus 120572

119899(119910)

1003816100381610038161003816

ge1003816100381610038161003816120572119899

(119910 + 1198940119902) minus 120572

119899(119910)

1003816100381610038161003816

1198940

otherwise(38)

Without the loss of generality we may assume that 119894max = 1198940+

1 which implies

sup119910le119909le119910+119898

119899119902

1003816100381610038161003816120572119899(119909) minus 120572

119899(119910)

1003816100381610038161003816 = max1le119894le119898

119899

1003816100381610038161003816120572119899(119910 + 119894119902) minus 120572

119899(119910)

1003816100381610038161003816

+1003816100381610038161003816120572119899

(1199090) minus 120572

119899(119910 + 119894

0119902)1003816100381610038161003816

(39)

Now applying inequality (36) the definition of 119894max and thetriangle inequality we have

sup119910le119909le119910+119898

119899119902

1003816100381610038161003816120572119899(119909) minus 120572

119899(119910)

1003816100381610038161003816

le max1le119894le119898

119899

1003816100381610038161003816120572119899(119910 + 119894119902) minus 120572

119899(119910)

1003816100381610038161003816

+1003816100381610038161003816120572119899

(119910 + [1198940+ 1] 119902) minus 120572

119899(119910 + 119894

0119902)1003816100381610038161003816

+ 119872radic119899119902 + 119862radic119899ℎ4

119899

le max1le119894le119898

119899

1003816100381610038161003816120572119899(119910 + 119894119902) minus 120572

119899(119910)

1003816100381610038161003816

+1003816100381610038161003816120572119899

(119910 + [1198940+ 1] 119902) minus 120572

119899(119910)

1003816100381610038161003816

+1003816100381610038161003816120572119899

(119910 + 1198940119902) minus 120572

119899(119910)

1003816100381610038161003816 + 119872radic119899119902 + 119862radic119899ℎ4

119899

le 3 max1le119894le119898

119899

1003816100381610038161003816120572119899(119910 + 119894119902) minus 120572

119899(119910)

1003816100381610038161003816 + 119872radic119899119902 + 119862radic119899ℎ4

119899

(40)

If we plug in 119902 = 119903119899we arrive at

sup119910le119909le119910+119898

119899119903119899

1003816100381610038161003816120572119899(119909) minus 120572

119899(119910)

1003816100381610038161003816 le 3 max1le119894le119898

119899

1003816100381610038161003816120572119899(119910 + 119894119903

119899) minus 120572

119899(119910)

1003816100381610038161003816

+ 119872radic119899119903119899+ 119862radic119899ℎ4

119899

(41)

Step 5 Finally we are in a position to obtain inequality (23)that is

forall120598gt0forall

120578gt0exist

0le120575lt12exist

1198990

forall119899ge1198990

119875( sup0le119910le1minus2120575

sup119910le119909le119910+2120575

1003816100381610038161003816120572119899(119909)minus120572

119899(119910)

1003816100381610038161003816

ge 120598) le 120578

(42)

We now successivelymake use the inequalities (29) (41) (27)(29) and (28) to get

119875( sup119910le119909le119910+2120575

1003816100381610038161003816120572119899(119909) minus 120572

119899(119910)

1003816100381610038161003816 ge 4119872120598 + 119862radic119899ℎ4

119899)

le 119875( sup119910le119909le119910+2(119898

119899+1)119903119899

1003816100381610038161003816120572119899(119909) minus 120572

119899(119910)

1003816100381610038161003816 ge 4119872120598 + 119862radic119899ℎ4

119899)

le 119875(3 max1le119894le2(119898119899+1)

1003816100381610038161003816120572119899(119910 + 119894119903

119899) minus 120572

119899(119910)

1003816100381610038161003816

+119872radic119899119903119899+ 119862radic119899ℎ4

119899ge 4119872120598 + 119862radic119899ℎ4

119899)

ISRN Probability and Statistics 5

= 119875( max1le119894le2(119898

119899+1)

1003816100381610038161003816120572119899(119910 + 119894119903

119899) minus 120572

119899(119910)

1003816100381610038161003816 ge 119872120598)

le

119862119901min119901

11199012+1199013

(119872120598)119901

(2120575)min119901

11199012+1199013le 120578

(43)

for any fixed 119910 isin [0 1] Since the upper bound does notdepend on 119910 and the probability measure as well as supre-mum function are continuous we obtain

119875( sup0le119910le1minus2120575

sup119910le119909le119910+2120575

1003816100381610038161003816120572119899(119909) minus 120572

119899(119910)

1003816100381610038161003816 ge 4119872120598 + 119862radic119899ℎ4

119899) le 120578

(44)

where 4119872120598 + 119862radic119899ℎ4

119899is arbitrarily small

Since condition (10) is checked the proof is completed

3 Tightness of the Standard Empirical Process

In this section we deal with the standard empirical processbuilt on an associated sequence of uniformly [0 1]distributedrvrsquos 119880

119895119895ge1

that is

120572119899(119909) =

1

radic119899

119899

sum

119895=1

(119868 [119880119895le 119909] minus 119909) (45)

where 119909 isin [0 1] We shall relax the restrictions imposedon the process by Yu in [7] to obtain tightness Preciselywe do not need stationarity any more due to the techniquedrawn from [14] and we lower the assumed rate at which thecovariance tends to zero

While proving tightness of the empirical process we willuse the criterion proved in the first section as well as some ofour covariance inequalities

We shall start with the fact known under the name ofmultinomial theorem We recall it in the following lemma

Lemma 2 For natural numbers119898 119899 and 1199091 119909

2 119909

119898isin R

(1199091+ sdot sdot sdot + 119909

119899)119898

= sum

1198961+sdotsdotsdot+119896

119899=119898

(119898

1198961 119896

119899

) prod

1le119894le119899

119909119896119894

119894 (46)

where (119898

1198961 119896

119899) = 119898(119896

1 sdot sdot sdot 119896

119899) and 119896

1 119896

119899isin

0 1 119898

In particular

(1199091+ sdot sdot sdot + 119909

119899)4

le 4 sdot sum

1198961+sdotsdotsdot+119896

119899=4

1199091198961

1sdot sdot sdot 119909

119896119899

119899 (47)

which implies for 119878119899= sum

119899

119894=1119883

119894 that

1198641198784

119899le 4 sdot sum

1198961+sdotsdotsdot+119896

119899=4

119864 (1198831198961

1sdot sdot sdot 119883

119896119899

119899)

le 4 sdot

119899

sum

119905=1

sum

0le119894119895119896le119899minus1

10038161003816100381610038161003816119864 (119883

119905119883

119905+119894119883

119905+119894+119895119883

119905+119894+119895+119896)10038161003816100381610038161003816

(48)

Let us now introduce the following notation due to Doukhanand Louhichi (see [14]) for a sequence of centered rvrsquos119883

119905119899

119899ge1

119862119903119902

= sup100381610038161003816100381610038161003816Cov (119883

1199051

sdot sdot sdot 119883119905119898

119883119905119898+119903sdot sdot sdot 119883

119905119902

)100381610038161003816100381610038161003816 (49)

where the supremum runs over all divisions of the groupcomposed of 119902 rvrsquos into two subgroups such that the distancebetween the highest index of the rvrsquos in the first group and thelowest index of the rvrsquos from the second group is equal to 119903119903 isin 1 2 For 119903 = 0 we shall define 119862

0119902= 1 Let us then

put

1198831199051

= 119868 [1198801199051

le 119909] minus 119909 1198831199052

= 119868 [1198801199052

le 119909] minus 119909 (50)

We will now estimate the summands in (48)If max119894 119895 119896 = 119894 then

10038161003816100381610038161003816119864 119883

119905sdot (119883

119905+119894119883

119905+119894+119895119883

119905+119894+119895+119896)10038161003816100381610038161003816

=100381610038161003816100381610038161003816Cov (119883

119905 119883

119905+119894119883

119905+119894+119895119883

119905+119894+119895+119896)100381610038161003816100381610038161003816le 119862

1198944

(51)

Similarly for max119894 119895 119896 = 119896 we have

10038161003816100381610038161003816119864 (119883

119905119883

119905+119894119883

119905+119894+119895) sdot 119883

119905+119894+119895+11989610038161003816100381610038161003816

=10038161003816100381610038161003816Cov (119883

119905119883

119905+119894119883

119905+119894+119895 119883

119905+119894+119895+119896)10038161003816100381610038161003816le 119862

1198964

(52)

When max119894 119895 119896 = 119895 then

10038161003816100381610038161003816119864 (119883

119905119883

119905+119894) sdot (119883

119905+119894+119895119883

119905+119894+119895+119896)10038161003816100381610038161003816

=10038161003816100381610038161003816Cov (119883

119905119883

119905+119894 119883

119905+119894+119895119883

119905+119894+119895+119896)

+119864 (119883119905119883

119905+119894) 119864 (119883

119905+119894+119895119883

119905+119894+119895+119896)10038161003816100381610038161003816

le 1198621198954

+ 1198621198942119862

1198962

(53)

We keep on estimating the fourth moment of 119878119899by intro-

ducing 119905 isin 0 119899mdashthe index for which |119864(119883119905119883

119905+119894119883

119905+119894+119895

119883119905+119894+119895+119896

)| attains its maximum

1198641198784

119899le 4119899 sum

0le119894119895119896le119899minus1

10038161003816100381610038161003816119864 (119883

119905119883

119905+119894119883

119905+119894+119895119883

119905+119894+119895+119896)10038161003816100381610038161003816

=4119899( sum

0le119894119896le119895

[1198621198942sdot 119862

1198962+ 119862

1198954]+ sum

0le119894119895le119896

1198621198964+ sum

0le119896119895le119894

1198621198944)

= 4119899( sum

0le119894119896le119895

1198621198942sdot 119862

1198962+ 3 sum

0le119896119895le119894

1198621198944)

(54)

6 ISRN Probability and Statistics

The terms obtained in (54) may further be bounded fromabove in the following way

sum

0le119894119896le119895

1198621198942sdot 119862

1198962=

119899minus1

sum

119895=0

119895

sum

119894119896=0

1198621198942119862

1198962

le 119899

119899minus1

sum

119894119896=0

1198621198942119862

1198962= 119899(

119899minus1

sum

119894=0

1198621198942)

2

sum

0le119896119895le119894

1198621198944

=

119899minus1

sum

119894=0

119894

sum

119895119896=0

1198621198944

le

119899minus1

sum

119894=0

(119894 + 1) (119894 + 1) 1198621198944

=

119899minus1

sum

119894=0

(119894 + 1)2119862

1198944

(55)

We thus get the inequality

1198641198784

119899le 4([119899

119899minus1

sum

119894=0

1198621198942]

2

+ 3119899

119899minus1

sum

119894=0

(119894 + 1)2119862

1198944) (56)

We shall now focus on estimating 1198621199032

and 1198621199034 Since 119880

1199051

and 1198801199052

are associated uniformly [0 1] distributed rvrsquos 1198831199051

and 1198831199052

mdashas monotone functions of these rvrsquosmdashare associat-ed as well In order to bound

1198621199032

= sup0le1199051lt1199052le119899 1199052minus1199051=119903

Cov (1198831199051

1198831199052

) (57)

let us notice that from Schwarz inequality

Cov (1198831199051

1198831199052

) = 119864 (119868 [1198801199051

le 119909] 119868 [1198801199052

le 119909]) minus 1199092

le radic1199092 minus 1199092le 119909

(58)

On the other hand invoking inequalities from [13 15]

Cov (1198831199051

1198831199052

)

le sup119909isin[01]

[119875 (1198801199051

le 1199091198801199052

le 119909) minus 119875 (1198801199051

le 119909)119875 (1198801199052

le 119909)]

le 119862 sdot Cov13

(1198801199051

1198801199052

) for associated rvrsquos4 sdot Cov (119880

1199051

1198801199052

) for MTP2rvrsquos

(59)

As a consequence

1198621199032

le min 119909 119862 sdot Cov13

(1198801199051

1198801199052

) for associated rvrsquosmin 119909 4 sdot Cov (119880

1199051

1198801199052

) for MTP2rvrsquos

(60)

where 1199052minus 119905

1= 119903 Still we need the upper bound for 119862

1199034 It

turns out that119862

1199034

= sup1003816100381610038161003816100381610038161003816100381610038161003816

Cov2

prod

119894=1

(119868 [119880119905119894

le 119909] minus 119909)

4

prod

119894=3

(119868 [119880119905119894

le 119909] minus 119909)

1003816100381610038161003816100381610038161003816100381610038161003816

(61)

In other words among all divisions of the group 119883119905119894

119894 isin

1 2 3 4 into two subgroups 1198621199034

attains the biggest valuein case we take two subgroups consisted of two rvrsquos Thesupremum runs over the set 119905

1 119905

2 119905

3 119905

4isin N 0 le 119905

1lt 119905

2lt

1199053lt 119905

4le 119899 and 119905

3minus 119905

2= 119903 Elementary calculation leads to the

following formula

1198621199034

= sup 10038161003816100381610038161003816119875 (119880119905119894

le 119909 119894 isin 1 2 3 4)

minus 119875 (119880119905119894

le 119909 119894 isin 1 2) 119875 (119880119905119894

le 119909 119894 isin 3 4)

minus 119909 [119875 (119880119905119894

le 119909 119894 isin 1 2 3)

minus 119875 (119880119905119894

le 119909 119894 isin 1 2) 119875 (1198801199053

le 119909)

+ 119875 (119880119905119894

le 119909 119894 isin 1 3 4)

minus 119875 (119880119905119894

le 119909 119894 isin 3 4) 119875 (1198801199051

le 119909)

+ 119875 (119880119905119894

le 119909 119894 isin 1 2 4)

minus 119875 (119880119905119894

le 119909 119894 isin 1 2) 119875 (1198801199054

le 119909)

+ 119875 (119880119905119894

le 119909 119894 isin 2 3 4)

minus119875 (119880119905119894

le 119909 119894 isin 3 4) 119875 (1198801199052

le 119909)]

+ 1199092[119875 (119880

1199051

le 1199091198801199053

le 119909)

minus 119875 (1198801199051

le 119909)119875 (1198801199053

le 119909)

+ 119875 (1198801199051

le 1199091198801199054

le 119909)

minus 119875 (1198801199051

le 119909)119875 (1198801199054

le 119909)

+ 119875 (1198801199052

le 1199091198801199053

le 119909)

minus 119875 (1198801199052

le 119909)119875 (1198801199053

le 119909)

+ 119875 (1198801199052

le 1199091198801199054

le 119909)

minus119875 (1198801199052

le 119909)119875 (1198801199052

le 119909)]10038161003816100381610038161003816

le10038161003816100381610038161198771

1003816100381610038161003816 +10038161003816100381610038161198772

1003816100381610038161003816 +10038161003816100381610038161198773

1003816100381610038161003816

(62)

where for the sake of simplicity 1198771 119877

2 and 119877

3are respec-

tively the free coefficient the expression with 119909 and theexpression with 119909

2 Using the Lebowitz inequality (see [15]for instance) and inequalities obtained in [13 15] we arrive at10038161003816100381610038161198771

1003816100381610038161003816 =10038161003816100381610038161003816119875 (119880

119905119894

le 119909 119894 isin 1 2 3 4)

minus119875 (119880119905119894

le 119909 119894 isin 1 2) 119875 (119880119905119894

le 119909 119894 isin 3 4)10038161003816100381610038161003816

le 11986711988011990511198801199053

+ 11986711988011990511198801199054

+ 11986711988011990521198801199053

+ 11986711988011990521198801199054

ISRN Probability and Statistics 7

le

119862(

2

sum

119894=1

4

sum

119895=3

Cov13(119880

119905119894

119880119905119895

)) for associated rvrsquos

4(

2

sum

119894=1

4

sum

119895=3

Cov (119880119905119894

119880119905119895

)) for MTP2rvrsquos

(63)

Analogously we get

10038161003816100381610038161198772

1003816100381610038161003816

le

2119909 sdot 119862(

2

sum

119894=1

4

sum

119895=3

Cov13(119880

119905119894

119880119905119895

)) for associated rvrsquos

2119909 sdot 4(

2

sum

119894=1

4

sum

119895=3

Cov (119880119905119894

119880119905119895

)) for MTP2rvrsquos

10038161003816100381610038161198773

1003816100381610038161003816

le

1199092sdot 119862(

2

sum

119894=1

4

sum

119895=3

Cov13(119880

119905119894

119880119905119895

) ) for associated rvrsquos

1199092sdot 4(

2

sum

119894=1

4

sum

119895=3

Cov (119880119905119894

119880119905119895

)) for MTP2rvrsquos

(64)

Eventually we have

1198621199034

le

119891 (119909) sdot119862(

2

sum

119894=1

4

sum

119895=3

Cov13(119880

119905119894

119880119905119895

)) for associated rvrsquos

119891 (119909) sdot4(

2

sum

119894=1

4

sum

119895=3

Cov (119880119905119894

119880119905119895

)) for MTP2rvrsquos

(65)

where 1199053minus 119905

2= 119903 and 119891(119909) = (1 + 2119909 + 119909

2) for 119909 isin [0 1]

Let us now introduce the following notation

120579119903=

sup119905isinN

Cov (119880119905 119880

119905+119903) for 119903 isin 1 2

1 for 119903 = 0

(66)

and assume it decays powerly at rate 119886 in the following way

120579119903= 119874(

1

(119903 + 1)119886) for 119886 gt 0 119903 isin 0 1 (67)

Let us get back to inequality (56) we can now carry on Atfirst for associated rvrsquos

1198641198784

119899le 4[119899

119899minus1

sum

119903=0

min 119909 119862 sdot Cov13(119880

1199051

1198801199052

)]

2

+ 4 sdot 3119899

119899minus1

sum

119903=0

(119903 + 1)2119891 (119909) sdot 119862(

2

sum

119894=1

4

sum

119895=3

Cov13(119880

119905119894

119880119905119895

))

le 119863 sdot ([119899

119899minus1

sum

119903=0

min 119909 (119903 + 1)minus1198863

]

2

+119899

119899minus1

sum

119903=0

(119903 + 1)2(119903 + 1)

minus1198863)

= 119863 sdot ([119899

119899

sum

119903=1

min 119909 119903minus1198863]

2

+ 119899

119899

sum

119903=1

1199032119903minus1198863

)

le 119863 sdot ([

[

119899 sum

119903lt119909minus3119886

119909 + 119899 sum

119903ge119909minus3119886

1

1199031198863]

]

2

+ 119899

119899

sum

119903=1

1199032minus1198863

)

le 1198631sdot (119899

2119909

2(119886minus3)119886+ 120585)

(68)

where119863 and1198631are constants and

120585 = 119899

119899

sum

119903=1

1199032minus1198863

=

119874(119899) 119886 gt 9

119874 (119899 ln 119899) 119886 = 9

119874 (1198994minus1198863

) 3 lt 119886 lt 9

(69)

It is worth mentioning that in the last inequality of (68) weused the estimate

int

infin

119909

1

119905119901119889119905 sim

1

119909119901minus1for 119901 gt 1 (70)

At the same time in the case of MTP2rvrsquos we get

1198641198784

119899le 119863

2sdot (119899

2119909

2(119886minus1)119886+ 120577) (71)

where1198632is constant and

120577 = 119899

119899

sum

119903=1

1199032minus119886

=

119874(119899) 119886 gt 3

119874 (119899 ln 119899) 119886 = 3

119874 (1198994minus119886

) 1 lt 119886 lt 3

(72)

Let now 120572119899(119909) = (1radic119899)sum

119899

119894=1(119868[119880

119905119894

le 119909] minus 119909) Then

120572119899(119909) minus 120572

119899(119910) =

1

radic119899

119899

sum

119894=1

(119868 [119909 lt 119880119905119894

le 119910] minus (119909 minus 119910))

for 119909 119910 isin [0 1]

(73)

8 ISRN Probability and Statistics

has the fourth moment estimatedmdashin the case of associatedrvrsquosmdashby

119864[120572119899(119909) minus 120572

119899(119910)]

4

= 119864[1

radic119899

119899

sum

119894=1

(119868 [119909 lt 119880119905119894

le 119910] minus (119909 minus 119910))]

4

=1

1198992119864[

119899

sum

119894=1

(119868 [119909 lt 119880119905119894

le 119910] minus (119909 minus 119910))]

4

le 1198631sdot (

1

119899211989921003816100381610038161003816119909 minus 119910

1003816100381610038161003816

2(119886minus3)119886

+120585

1198992)

= 1198631sdot (

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

2(119886minus3)119886

+120585

1198992)

=

119874(119899minus1) 119886 gt 9

119874(ln 119899119899

) 119886 = 9

119874 (1198992minus1198863

) 3 lt 119886 lt 9

(74)

and in the case of MTP2rvrsquos by

119864[120572119899(119909) minus 120572

119899(119910)]

4

le 1198632sdot (

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

2(119886minus1)119886

+120577

1198992)

=

119874(119899minus1) 119886 gt 3

119874(ln 119899119899

) 119886 = 3

119874 (1198992minus119886

) 1 lt 119886 lt 3

(75)

In light of the Shao and Yursquos criterion our process is tight forassociated rvrsquos when 119886 gt 6 and for MTP

2rvrsquos when 119886 gt 2

Let us sum up this result in the following theorem

Theorem 3 Let 120572119899(119909) = (1radic119899)sum

119899

119894=1(119868[119880

119894le 119909] minus 119909) be the

empirical process built on an associated sequence of uniformly[0 1] distributed rvrsquos 119880

119894119894ge1

Let also

120579119903= sup

119905isinNCov (119880

119905 119880

119905+119903) = 119874 ((119903 + 1)

minus119886)

where 119903 isin 0 1 119886 gt 0

(76)

Then 120572119899(119909)

119899ge1is tight for 119886 gt 6 If the rvrsquos 119880

119894119894ge1

are MTP2

then the process is tight for 119886 gt 2

Yu assumed stationarity of 119880119894119894ge1

andsum

infin

119899=111989965+]Cov(119880

0 119880

119899) lt infin for a positive constant ]

thus the rate of decay 119886 gt 7 5 Our result weakens consid-erably these assumptions especially in the case of MTP

2rvrsquos

Louhichi in [16] proposed a different tightness criterioninvolving the so-called bracketing numbers She managedto enhance Yursquos resultmdasheven more than Shao and Yu in[8]mdashsince she proved that it suffices to take 119886 gt 4 to gettightness of the empirical process based on the associated

rvrsquos Nevertheless she kept the assumption of stationarityvalid

In the final analysis our resultrsquos advantage is the absenceof the stationarity assumption and the rate of decay for120579119903remains (up to the authorrsquos knowledge) unimproved for

MTP2rvrsquos

Unfortunately with a view to obtainingweak convergenceof the process in question that is also convergence offinite-dimensional distributions we do not know how tomanage without the assumption of stationarityTherefore weconclude with the following corollary

Corollary 4 Let 120572119899(119909) = (1radic119899)sum

119899

119894=1(119868[119880

119894le 119909] minus 119909) be the

empirical process built on a stationary associated sequence ofuniformly [0 1] distributed rvrsquos 119880

119894119894ge1

Let also

120579119903= Cov (119880

1 119880

1+119903) = 119874 ((119903 + 1)

minus119886)

where 119903 isin 0 1 119886 gt 0

(77)

Then if 119886 gt 6

120572119899(sdot) 997888rarr 119861 (sdot) weakly in 119863 [0 1] (78)

where 119861(sdot) is the zero mean Gaussian process on [0 1] withcovariance structure defined by

1205902(119909 119910) = 119909 and 119910 minus 119909119910

+

infin

sum

119895=1

[ Cov (119868 [1198801le 119909] 119868 [119880

119895+1le 119910])

+ Cov (119868 [1198801le 119910] 119868 [119880

119895+1le 119909])]

(79)

If the rvrsquos 119880119894119894ge1

are MTP2 then it suffices to be 119886 gt 2 in

order to claim the above convergence

Proof It remains to establish convergence of finite-dimen-sional distributions repeating the procedure from [7]

4 Tightness of the Kernel-TypeEmpirical Process

In this section we shall weaken assumption imposed on thecovariance structure of rvrsquos 119883

119895119895ge1

by Cai and Roussas in [1]for the kernel estimator of the df

119865119899(119909) =

1

119899

119899

sum

119895=1

119870(

119909 minus 119883119895

ℎ119899

) (80)

They deal with a stationary sequence of negatively associatedrvrsquos (cf [17]) and need the same condition as Yu [7] that is

1003816100381610038161003816Cov (1198831 119883

1+119899)1003816100381610038161003816 = 119874(

1

11989975+]) (81)

to get tightness of the smooth empirical process (see condi-tion (A4) in [1])

ISRN Probability and Statistics 9

It turns out that it suffices to have

1003816100381610038161003816Cov (1198801 119880

1+119899)1003816100381610038161003816 = 119874(

1

1198993119901(119901minus1)) (82)

where 119901 gt 4 is a positive constant taken from the tightnesscriterion (8) It is easy to see that asymptotically we get therate 3

On the way to prove it we will also take use of aRosenthal-type inequality due to Shao andYu (seeTheorem 2in [8]) we shall recall in the following lemma

Lemma 5 Let 119901 gt 2 and 119891 be a real valued function boundedby 1 with bounded first derivative Suppose that 119883

119899119899ge1

is asequence of stationary and associated rvrsquos such that for 119899 isin N

Cov (1198831 119883

119899) = 119874 (119899

minus119887) for some 119887 gt 119901 minus 1 (83)

Then for any 120583 gt 0 there exists some positive constant 119896120583

independent of the function 119891 for which

1198641003816100381610038161003816119878119899

(119891) minus 119864119878119899(119891)

1003816100381610038161003816

119901

le 119896120583(119899

1+12058310038171003817100381710038171003817119891

101584010038171003817100381710038171003817

2

+1198991199012 [

[

119899

sum

119895=1

10038161003816100381610038161003816Cov (119891 (119883

1)119891 (119883

119895))10038161003816100381610038161003816]

]

)

(84)

As we can see the lemma assumes association but itworks for negatively associated rvrsquos as well since in the proofit reaches back the result of Newman (see Proposition 15 in[18]) where both types of association are allowed

Let us recall that 120572119899(119909) = radic119899(119865

119899(119909) minus 119864119865

119899(119909)) where

119865119899(119909) = (1119899)sum

119899

119895=1119870((119909 minus 119883

119895)ℎ

119899) It is easy to see that

1198641003816100381610038161003816120572119899

(119909) minus 120572119899(119910)

1003816100381610038161003816

119901

= 119899minus1199012

11986410038161003816100381610038161003816119878119899(119891 (119883

119895)) minus 119864119878

119899(119891 (119883

119895))10038161003816100381610038161003816

119901

(85)

where

119891 (119883119895) = 119870(

119909 minus 119883119895

ℎ119899

) minus 119870(

119910 minus 119883119895

ℎ119899

)

119878119899(119891) =

119899

sum

119895=1

119891 (119883119895)

(86)

With an intent to use Lemma 5 we need 119891 to be boundedfrom above by 1 (which in our case is obvious) and to havebounded 119891

1015840 thus we assume

sup119905isinR

10038161003816100381610038161003816119870

1015840(119905)

10038161003816100381610038161003816

1

ℎ119899

le119862

119870

ℎ119899

lt infin (87)

Now applying inequality (84) we have

1198641003816100381610038161003816120572119899

(119909) minus 120572119899(119910)

1003816100381610038161003816

119901

le119896120583119899minus1199012

1198991+120583

1198622

119870

ℎ2

119899

+1198991199012[

[

119899

sum

119895=1

10038161003816100381610038161003816Cov (119891 (119883

1)119891 (119883

119895))10038161003816100381610038161003816]

]

1199012

(88)

Newman showed in [18] that

Cov (1198911(119883) 119891

2(119884))

= ∬R2119891

1015840

1(119909) 119891

1015840

2(119910)

times [119875 (119883 le 119909 119884 le 119910) minus 119875 (119883 le 119909) 119875 (119884 le 119910)] 119889119909 119889119910

(89)

if 1198911and 119891

2are real valued functions on R having square

integrable derivatives 1198911015840

1and 119891

1015840

2 respectively and provided

that 1198911(119883) 119891

2(119884) have finite secondmoments In light of that

equation and linearity of covariance we get10038161003816100381610038161003816Cov (119891 (119883

1) 119891 (119883

119895))10038161003816100381610038161003816

=

1003816100381610038161003816100381610038161003816∬

R2[119875 (119883

1lt 119909 minus ℎ

119899119904 119883

119895lt 119909 minus ℎ

119899119905)

minus119875 (1198831lt 119909 minus ℎ

119899119904 119883

119895lt 119910 minus ℎ

119899119905)]

minus [119875 (1198831lt 119909 minus ℎ

119899119904) 119875 (119883

119895lt 119909 minus ℎ

119899119905)

minus119875 (1198831lt 119909 minus ℎ

119899119904) 119875 (119883

119895lt 119910 minus ℎ

119899119905)]

minus [119875 (1198831lt 119910 minus ℎ

119899119904 119883

119895lt 119909 minus ℎ

119899119905)

minus119875 (1198831lt 119910 minus ℎ

119899119904 119883

119895lt 119910 minus ℎ

119899119905)]

minus [119875 (1198831lt 119910 minus ℎ

119899119904) 119875 (119883

119895lt 119910 minus ℎ

119899119905)

minus 119875 (1198831lt 119910 minus ℎ

119899119904)

times 119875 (119883119895lt 119909 minus ℎ

119899119905)] 119896 (119904) 119896 (119905) 119889119904 119889119905

10038161003816100381610038161003816100381610038161003816

(90)

Without the loss of generality we may and do assume that119909 gt 119910 thus10038161003816100381610038161003816Cov (119891 (119883

1) 119891 (119883

119895))10038161003816100381610038161003816

=

1003816100381610038161003816100381610038161003816∬

R2119875 (119910 minus ℎ

119899119904 le 119883

1lt 119909 minus ℎ

119899119904

119910 minus ℎ119899119905 le 119883

119895lt 119909 minus ℎ

119899119905)

minus 119875 (119910 minus ℎ119899119904 le 119883

1lt 119909 minus ℎ

119899119904)

times 119875 (119910 minus ℎ119899119905 le 119883

119895lt 119909 minus ℎ

119899119905) 119896 (119904) 119896 (119905) 119889119904 119889119905

1003816100381610038161003816100381610038161003816

(91)

and by triangle inequality we get10038161003816100381610038161003816Cov (119891 (119883

1) 119891 (119883

119895))10038161003816100381610038161003816

le ∬R2[int

119909minusℎ119899119904

119910minusℎ119899119904

int

119909minusℎ119899119905

119910minusℎ119899119905

1198911198831119883119895(119906 V) 119889119906 119889V

+int

119909minusℎ119899119904

119910minusℎ119899119904

1198911198831(119906) 119889119906int

119909minusℎ119899119905

119910minusℎ119899119905

119891119883119895(V) 119889V] 119896 (119904) 119896 (119905) 119889119904 119889119905

(92)

10 ISRN Probability and Statistics

1198911198831119883119895

(119906 V) 1198911198831

(119906) and 119891119883119895

(V) are the joint pdf of [1198831 119883

119895]

and marginal pdfrsquos of rvrsquos1198831and119883

119895 respectively We need

to further assume that 1198621stands for a common upper bound

of 1198911198831

and 1198911198831119883119895

for all 119895 isin N that is

100381710038171003817100381710038171198911198831

10038171003817100381710038171003817le 119862

1

1003817100381710038171003817100381710038171198911198831119883119895

100381710038171003817100381710038171003817le 119862

1forall119895 ge 1 (93)

Then we finally obtain

10038161003816100381610038161003816Cov (119891 (119883

1) 119891 (119883

119895))10038161003816100381610038161003816le 2119862

1

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

2

where = max 1198621 119862

2

1

(94)

On the other hand proceeding like Cai and Roussas in [1] wecan shortly get

10038161003816100381610038161003816Cov (119891 (119883

1) 119891 (119883

119895))10038161003816100381610038161003816

le

100381610038161003816100381610038161003816100381610038161003816

Cov(119870(119909 minus 119883

1

ℎ119899

) 119870(

119909 minus 119883119895

ℎ119899

))

100381610038161003816100381610038161003816100381610038161003816

+

100381610038161003816100381610038161003816100381610038161003816

Cov(119870(119909 minus 119883

1

ℎ119899

) 119870(

119910 minus 119883119895

ℎ119899

))

100381610038161003816100381610038161003816100381610038161003816

+

100381610038161003816100381610038161003816100381610038161003816

Cov(119870(119910 minus 119883

1

ℎ119899

) 119870(

119909 minus 119883119895

ℎ119899

))

100381610038161003816100381610038161003816100381610038161003816

+

100381610038161003816100381610038161003816100381610038161003816

Cov(119870(119910 minus 119883

1

ℎ119899

) 119870(

119910 minus 119883119895

ℎ119899

))

100381610038161003816100381610038161003816100381610038161003816

le 41198622

10038161003816100381610038161003816Cov (119883

1 119883

119895)10038161003816100381610038161003816

13

(95)

where 1198622is a constant relevant to the covariance inequality

for negatively associated rvrsquos (see [13])We now arrive at the following inequality

10038161003816100381610038161003816Cov (119891 (119883

1) 119891 (119883

119895))10038161003816100381610038161003816

le max 1198621 119862

2

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

2

10038161003816100381610038161003816Cov(119883

1 119883

119895)10038161003816100381610038161003816

13

(96)

Assuming |Cov(119891(1198831) 119891(119883

119895))| = 119874(119895

minus119903) and proceeding

similarly to (68) we can get

[

[

119899

sum

119895=1

10038161003816100381610038161003816Cov (119891 (119883

1) 119891 (119883

119895))10038161003816100381610038161003816]

]

1199012

le max 1198621 119862

2 [

[

sum

119895lt|119909minus119910|minus2(1199033)

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

2

+ sum

119895ge|119909minus119910|minus2(1199033)

1

1198951199033]

]

1199012

le 2max 1198621 119862

21003816100381610038161003816119909 minus 119910

1003816100381610038161003816

((119903minus3)119903)119901

(97)

To sum up we obtain the following inequality

1198641003816100381610038161003816120572119899

(119909) minus 120572119899(119910)

1003816100381610038161003816

119901

le 119896120583119899

1+120583minus1199012119862

2

119870

ℎ2

119899

+ 2max 1198621 119862

21003816100381610038161003816119909 minus 119910

1003816100381610038161003816

((119903minus3)119903)119901

le 1198621198991+120583minus1199012 1

ℎ2

119899

+1003816100381610038161003816119909 minus 119910

1003816100381610038161003816

((119903minus3)119903)119901

(98)

where 119862 = 119896120583max1198622

119896 2max119862

1 119862

2 The formula of the

tightness criterion (8) implies that ((119903 minus 3)119903)119901 gt 1 so

119903 gt3119901

119901 minus 1 where 119901 gt 2 (99)

Let us recall the assumptions imposed on the bandwidhtsℎ

119899119899ge1

by Cai and Roussas in [1]B1 lim

119899rarrinfinℎ119899= 0 and ℎ

119899gt 0 for all 119899 isin N

+

B2 lim119899rarrinfin

119899ℎ119899= infin thus ℎ

119899= 119874(1119899

1minus120573) 120573 gt 0

B3 lim119899rarrinfin

119899ℎ4

119899= 0 hence ℎ

119899= 119874(1119899

14+120575) 120575 gt 0

In light of the above

1198991+120583minus1199012 1

ℎ2

119899

= 11989932+120583+2120575minus1199012

(100)

where 120583 gt 0 119901 gt 2 and 0 lt 120575 lt 34 Confrontation with thetightness criterion (8) forces

3

2+ 120583 + 2120575 minus

119901

2lt minus

1

2 (101)

which implies 119901 gt 4 Let us conclude with the followingtheorem

Theorem6 Let 120572119899(119909) = radic119899(119865

119899(119909)minus119864119865

119899(119909)) be the empirical

process built on the kernel estimator of the df119865 for a stationarysequence of negatively associated rvrsquos Assume that conditionsA1 A2 (87) B1 B2 B3 are satisfied Then provided that thetightness criterion (8) holds with 119901 gt 4 it suffices to demand

1003816100381610038161003816Cov (1198801 119880

1+119899)1003816100381610038161003816 = 119874(

1

1198993119901(119901minus1)) (102)

in order to obtain120572

119899(sdot) 997888rarr 119861 (sdot) weakly in 119863 [0 1] (103)

where 119861(sdot) is the zero mean Gaussian process with covariancestructure defined by

1205902(119909 119910) = 119865 (119909 and 119910) minus 119865 (119909) 119865 (119910)

+

infin

sum

119895=1

[ Cov (119868 [1198831le 119909] 119868 [119883

119895+1le 119910])

+ Cov (119868 [1198831le 119910] 119868 [119883

119895+1le 119909])]

(104)

and 119909 119910 isin [0 1]

Proof The remaining covergence of finite-dimensional dis-tributions of 120572

119899(119909) is established in [1]

ISRN Probability and Statistics 11

5 Weak Convergence of theRecursive Kernel-Type Empirical Processunder IID Assumption

The aim of this section is to show weak convergence of theempirical process

120572119899(119909) =

1

radic119899

119899

sum

119895=1

[119870(

119909 minus 119883119895

ℎ119895

) minus 119864119870(

119909 minus 119883119895

ℎ119895

)] (105)

built on iid rvrsquos 119883119895119895ge1

Wewill prove that120572

119899(sdot) convergesweakly to the Brownian

bridge 119861(sdot) with the following covariance structure

1205902(119909 119910) = 119865 (119909 and 119910) minus 119865 (119909) 119865 (119910) for 119909 119910 isin [0 1]

(106)

It turns out that the tightness criterion (8) suffices to reachthe goal Convergence of finite-dimensional distributions of120572

119899(sdot) holds and we shall show itProceeding like Cai and Roussas in [1] we need to show

that for any 119886 119887 isin R

119886120572119899(119909) + 119887120572

119899(119910) 997888rarr 119886119861 (119909) + 119887119861 (119910) in distribution

(107)

Let us introduce the notation

119884119894(119909) = 119870(

119909 minus 119883119894

ℎ119894

) minus 119864119870(119909 minus 119883

119894

ℎ119894

) (108)

and look into the covariance structure of 120572119899(sdot)

Cov (120572119899(119909) 120572

119899(119910))

= Cov( 1

radic119899

119899

sum

119894=1

119884119894(119909)

1

radic119899

119899

sum

119894=1

119884119894(119910))

=1

119899

119899

sum

119894=1

Cov (119884119894(119909) 119884

119894(119910))

=1

119899

119899

sum

119894=1

Cov(119870(119909 minus 119883

119894

ℎ119894

) 119870(119910 minus 119883

119894

ℎ119894

))

(109)

where the second equality follows from assumed indepen-dence of rvrsquos 119883

119894119894ge1

Firstly we observe that each summand converges to119865(119909and

119910) minus 119865(119909)119865(119910) since

Cov(119870(119909 minus 119883

119894

ℎ119894

) 119870(119910 minus 119883

119894

ℎ119894

))

= 119864[119870(119909 minus 119883

119894

ℎ119894

)119870(119910 minus 119883

119894

ℎ119894

)]

minus 119864119870(119909 minus 119883

119894

ℎ119894

)119864119870(119910 minus 119883

119894

ℎ119894

)

(110)

where

119864[119870(119909 minus 119883

119894

ℎ119894

)119870(119910 minus 119883

119894

ℎ119894

)]

= int

119909and119910

minusinfin

119870(119909 minus 119905

ℎ119894

)119870(119910 minus 119905

ℎ119894

)119889119865 (119905)

+ int

119909or119910

119909and119910

119870(119909 minus 119905

ℎ119894

)119870(119910 minus 119905

ℎ119894

)119889119865 (119905)

+ int

infin

119909or119910

119870(119909 minus 119905

ℎ119894

)119870(119910 minus 119905

ℎ119894

)119889119865 (119905)

(111)

and 119865 denotes the common df of the rvrsquos 119883119894119894ge1

In [6] itwas shown that

119864119870(119909 minus 119883

119894

ℎ119894

) 997888rarr 119865 (119909) (112)

and recalling that the kernel function 119870 is a df as well wearrive at the conclusion

Secondly applying Toeplitz lemma we get

Cov (120572119899(119909) 120572

119899(119910)) 997888rarr 119865 (119909 and 119910) minus 119865 (119909) 119865 (119910)

= 1205902(119909 119910) 119899 997888rarr infin

(113)

Thus

Var (119886120572119899(119909) + 119887120572

119899(119910))

= 1198862 Var (120572

119899(119909)) + 2119886119887Cov (120572

119899(119909) 120572

119899(119910))

+ 1198872 Var (120572

119899(119910))

997888rarr 1198862120590

2(119909 119909) + 2119886119887120590

2(119909 119910) + 119887

2120590

2(119910 119910) 119899 997888rarr infin

(114)

We are now on the way to prove that 119886120572119899(119909) + 119887120572

119899(119910) con-

verges in distribution to 119886119861(119909) + 119887119861(119910) sim N(0 1198862120590

2(119909 119909) +

21198861198871205902(119909 119910) + 119887

2120590

2(119910 119910)) which in other words means that

1

radic119899

infin

sum

119894=1

119886119884119894(119909) + 119887119884

119894(119910)

radicVar (119886120572119899(119909) + 119887120572

119899(119910))

997888rarr N (0 1) (115)

In order to obtain (115) we will use Lyapunov condition

lim119899rarrinfin

1

1198632+120575

119899

119899

sum

119894=1

1198641003816100381610038161003816119886119884119894

(119909) + 119887119884119894(119910)

1003816100381610038161003816

2+120575

= 0 (116)

for 120575 = 1 where 1198632

119899= Var(sum119899

119894=1[119886119884

119894(119909) + 119887119884

119894(119910)]) Since

|119870(sdot) minus 119864119870(sdot)| le 2

1198641003816100381610038161003816119886119884119894

(119909) + 119887119884i (119910)1003816100381610038161003816

3

le 8 (1198863+ 119887

3) (117)

Moreover 1198632

119899= 119899Var(119886120572

119899(119909) + 119887120572

119899(119910)) together with (114)

yields

1198633

119899= 119874 (119899

32) (118)

12 ISRN Probability and Statistics

Combining (117) with (118) we obtain

1

1198633

119899

119899

sum

119894=1

1198641003816100381610038161003816119886119884119894

(119909) + 119887119884119894(119910)

1003816100381610038161003816

3

= 119874 (119899minus12

) (119)

which shows that Lyapunov condition holds and completesthe proof of convergence of finite-dimensional distributionsof 120572

119899(119909)We summarize the result of that section in the following

theorem

Theorem 7 Let 120572119899(119909) be the recursive kernel-type empirical

process defined by (105) built on iid rvrsquos 119883119895119895ge1

withmarginal df 119865 If 120572

119899(sdot) satisfies the tightness criterion (8) then

120572119899(sdot) 997888rarr 119861 (sdot) weakly in 119863 [0 1] (120)

where 119861 is the Brownian bridge

Acknowledgment

The author would like to thank the referees for the carefulreading of the paper and helpful remarks

References

[1] Z Cai and G G Roussas ldquoWeak convergence for smoothestimator of a distribution function under negative associationrdquoStochastic Analysis and Applications vol 17 no 2 pp 145ndash1681999

[2] Z Cai andG G Roussas ldquoEfficient Estimation of a DistributionFunction under Qudrant Dependencerdquo Scandinavian Journal ofStatistics vol 25 no 1 pp 211ndash224 1998

[3] ZW Cai and G G Roussas ldquoUniform strong estimation under120572-mixing with ratesrdquo Statistics amp Probability Letters vol 15 no1 pp 47ndash55 1992

[4] G G Roussas ldquoKernel estimates under association stronguniform consistencyrdquo Statistics amp Probability Letters vol 12 no5 pp 393ndash403 1991

[5] Y F Li and S C Yang ldquoUniformly asymptotic normalityof the smooth estimation of the distribution function underassociated samplesrdquo Acta Mathematicae Applicatae Sinica vol28 no 4 pp 639ndash651 2005

[6] Y Li C Wei and S Yang ldquoThe recursive kernel distributionfunction estimator based onnegatively and positively associatedsequencesrdquo Communications in Statistics vol 39 no 20 pp3585ndash3595 2010

[7] H Yu ldquoA Glivenko-Cantelli lemma and weak convergence forempirical processes of associated sequencesrdquo ProbabilityTheoryand Related Fields vol 95 no 3 pp 357ndash370 1993

[8] Q M Shao and H Yu ldquoWeak convergence for weightedempirical processes of dependent sequencesrdquo The Annals ofProbability vol 24 no 4 pp 2098ndash2127 1996

[9] P Billingsley Convergence of Probability Measures Wiley Seriesin Probability and Statistics Probability and Statistics JohnWiley amp Sons New York NY USA 1999

[10] J D Esary F Proschan and D W Walkup ldquoAssociation ofrandom variables with applicationsrdquo Annals of MathematicalStatistics vol 38 pp 1466ndash1474 1967

[11] A Bulinski and A Shashkin Limit Theorems for AssociatedRandom Fields and Related Systems vol 10 of Advanced Serieson Statistical Science and Applied Probability World Scientific2007

[12] S Karlin and Y Rinott ldquoClasses of orderings of measures andrelated correlation inequalities I Multivariate totally positivedistributionsrdquo Journal of Multivariate Analysis vol 10 no 4 pp467ndash498 1980

[13] P Matuła and M Ziemba ldquoGeneralized covariance inequali-tiesrdquo Central European Journal of Mathematics vol 9 no 2 pp281ndash293 2011

[14] P Doukhan and S Louhichi ldquoA new weak dependence condi-tion and applications to moment inequalitiesrdquo Stochastic Proc-esses and their Applications vol 84 no 2 pp 313ndash342 1999

[15] P Matuła ldquoA note on some inequalities for certain classes ofpositively dependent random variablesrdquo Probability and Math-ematical Statistics vol 24 no 1 pp 17ndash26 2004

[16] S Louhichi ldquoWeak convergence for empirical processes of asso-ciated sequencesrdquo Annales de lrsquoInstitut Henri Poincare vol 36no 5 pp 547ndash567 2000

[17] K Joag-Dev and F Proschan ldquoNegative association of randomvariables with applicationsrdquoThe Annals of Statistics vol 11 no1 pp 286ndash295 1983

[18] C M Newman ldquoAsymptotic independence and limit theoremsfor positively and negatively dependent random variablesrdquo inInequalities in Statistics and Probability Y L Tong Ed vol 5pp 127ndash140 Institute ofMathematical StatisticsHayward CalifUSA 1984

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 ISRN Probability and Statistics

Since 119870 is a nondecreasing function and applying triangleinequality again we get

1003816100381610038161003816120572119899(119909) minus 120572

119899(119910)

1003816100381610038161003816

le1003816100381610038161003816120572119899

(119910 + 119902) minus 120572119899(119910)

1003816100381610038161003816 +radic119899

10038161003816100381610038161003816119864119865

119899(119910 + 119902) minus 119864119865

119899(119910)

10038161003816100381610038161003816

+ radic119899 (10038161003816100381610038161003816119864119865

119899(119909) minus 119865 (119909)

10038161003816100381610038161003816+10038161003816100381610038161003816119864119865

119899(119910) minus 119865 (119910)

10038161003816100381610038161003816

+1003816100381610038161003816119865 (119909) minus 119865 (119910)

1003816100381610038161003816 )

(33)

Cai and Roussas in [1] showed that under assumptions madeon the df 119865 and the kernel function119870 we have

10038161003816100381610038161003816119864119865

119899(119909) minus 119865 (119909)

10038161003816100381610038161003816= 119874 (ℎ

2

119899) (34)

Similarly by Taylor expansion it is easy to see that

1003816100381610038161003816119865 (119909) minus 119865 (119910)1003816100381610038161003816 le

10038171003817100381710038171198911003817100381710038171003817 sdot

1003816100381610038161003816119909 minus 1199101003816100381610038161003816 (35)

Thus

1003816100381610038161003816120572119899(119909) minus 120572

119899(119910)

1003816100381610038161003816 le1003816100381610038161003816120572119899

(119910 + 119902) minus 120572119899(119910)

1003816100381610038161003816

+ 119862radic119899ℎ4

119899+119872radic119899119902

(36)

where 119872 = sup119909isin[01]

119891(119909) and 119862 = 1198621198911015840119870

is a positive con-stant dependent on the functions 119865 and119870

Let us recall that 119910 le 119909 le 119910+ 119902119898119899= lfloor120575119903

119899rfloor and 119902 ge 120598119899

Taking 119902 = (120598radic119899)(= 119903119899) we notice that 119898

119899119902 = lfloor120575119903

119899rfloor119903

119899le 120575

thus when 119899 rarr infin the interval [119910 119910 + 119898119899119902] coincides with

[119910 119910 + 120575] Approaching the aim of Step 4 let us observe that

sup119910le119909le119910+119898

119899119902

1003816100381610038161003816120572119899(119909) minus 120572

119899(119910)

1003816100381610038161003816 = max1le119894le119898

119899

1003816100381610038161003816120572119899(119910 + 119894119902) minus 120572

119899(119910)

1003816100381610038161003816

+1003816100381610038161003816120572119899

(1199090) minus 120572

119899(119910 + 119894max119902)

1003816100381610038161003816

(37)

where 1199090isin [119910+119894

0119902 119910+(119894

0+1)119902] is a point at which the above

supremum is attained and

119894max =

1198940+ 1 if 1003816100381610038161003816120572119899

(119910 + [1198940+ 1] 119902) minus 120572

119899(119910)

1003816100381610038161003816

ge1003816100381610038161003816120572119899

(119910 + 1198940119902) minus 120572

119899(119910)

1003816100381610038161003816

1198940

otherwise(38)

Without the loss of generality we may assume that 119894max = 1198940+

1 which implies

sup119910le119909le119910+119898

119899119902

1003816100381610038161003816120572119899(119909) minus 120572

119899(119910)

1003816100381610038161003816 = max1le119894le119898

119899

1003816100381610038161003816120572119899(119910 + 119894119902) minus 120572

119899(119910)

1003816100381610038161003816

+1003816100381610038161003816120572119899

(1199090) minus 120572

119899(119910 + 119894

0119902)1003816100381610038161003816

(39)

Now applying inequality (36) the definition of 119894max and thetriangle inequality we have

sup119910le119909le119910+119898

119899119902

1003816100381610038161003816120572119899(119909) minus 120572

119899(119910)

1003816100381610038161003816

le max1le119894le119898

119899

1003816100381610038161003816120572119899(119910 + 119894119902) minus 120572

119899(119910)

1003816100381610038161003816

+1003816100381610038161003816120572119899

(119910 + [1198940+ 1] 119902) minus 120572

119899(119910 + 119894

0119902)1003816100381610038161003816

+ 119872radic119899119902 + 119862radic119899ℎ4

119899

le max1le119894le119898

119899

1003816100381610038161003816120572119899(119910 + 119894119902) minus 120572

119899(119910)

1003816100381610038161003816

+1003816100381610038161003816120572119899

(119910 + [1198940+ 1] 119902) minus 120572

119899(119910)

1003816100381610038161003816

+1003816100381610038161003816120572119899

(119910 + 1198940119902) minus 120572

119899(119910)

1003816100381610038161003816 + 119872radic119899119902 + 119862radic119899ℎ4

119899

le 3 max1le119894le119898

119899

1003816100381610038161003816120572119899(119910 + 119894119902) minus 120572

119899(119910)

1003816100381610038161003816 + 119872radic119899119902 + 119862radic119899ℎ4

119899

(40)

If we plug in 119902 = 119903119899we arrive at

sup119910le119909le119910+119898

119899119903119899

1003816100381610038161003816120572119899(119909) minus 120572

119899(119910)

1003816100381610038161003816 le 3 max1le119894le119898

119899

1003816100381610038161003816120572119899(119910 + 119894119903

119899) minus 120572

119899(119910)

1003816100381610038161003816

+ 119872radic119899119903119899+ 119862radic119899ℎ4

119899

(41)

Step 5 Finally we are in a position to obtain inequality (23)that is

forall120598gt0forall

120578gt0exist

0le120575lt12exist

1198990

forall119899ge1198990

119875( sup0le119910le1minus2120575

sup119910le119909le119910+2120575

1003816100381610038161003816120572119899(119909)minus120572

119899(119910)

1003816100381610038161003816

ge 120598) le 120578

(42)

We now successivelymake use the inequalities (29) (41) (27)(29) and (28) to get

119875( sup119910le119909le119910+2120575

1003816100381610038161003816120572119899(119909) minus 120572

119899(119910)

1003816100381610038161003816 ge 4119872120598 + 119862radic119899ℎ4

119899)

le 119875( sup119910le119909le119910+2(119898

119899+1)119903119899

1003816100381610038161003816120572119899(119909) minus 120572

119899(119910)

1003816100381610038161003816 ge 4119872120598 + 119862radic119899ℎ4

119899)

le 119875(3 max1le119894le2(119898119899+1)

1003816100381610038161003816120572119899(119910 + 119894119903

119899) minus 120572

119899(119910)

1003816100381610038161003816

+119872radic119899119903119899+ 119862radic119899ℎ4

119899ge 4119872120598 + 119862radic119899ℎ4

119899)

ISRN Probability and Statistics 5

= 119875( max1le119894le2(119898

119899+1)

1003816100381610038161003816120572119899(119910 + 119894119903

119899) minus 120572

119899(119910)

1003816100381610038161003816 ge 119872120598)

le

119862119901min119901

11199012+1199013

(119872120598)119901

(2120575)min119901

11199012+1199013le 120578

(43)

for any fixed 119910 isin [0 1] Since the upper bound does notdepend on 119910 and the probability measure as well as supre-mum function are continuous we obtain

119875( sup0le119910le1minus2120575

sup119910le119909le119910+2120575

1003816100381610038161003816120572119899(119909) minus 120572

119899(119910)

1003816100381610038161003816 ge 4119872120598 + 119862radic119899ℎ4

119899) le 120578

(44)

where 4119872120598 + 119862radic119899ℎ4

119899is arbitrarily small

Since condition (10) is checked the proof is completed

3 Tightness of the Standard Empirical Process

In this section we deal with the standard empirical processbuilt on an associated sequence of uniformly [0 1]distributedrvrsquos 119880

119895119895ge1

that is

120572119899(119909) =

1

radic119899

119899

sum

119895=1

(119868 [119880119895le 119909] minus 119909) (45)

where 119909 isin [0 1] We shall relax the restrictions imposedon the process by Yu in [7] to obtain tightness Preciselywe do not need stationarity any more due to the techniquedrawn from [14] and we lower the assumed rate at which thecovariance tends to zero

While proving tightness of the empirical process we willuse the criterion proved in the first section as well as some ofour covariance inequalities

We shall start with the fact known under the name ofmultinomial theorem We recall it in the following lemma

Lemma 2 For natural numbers119898 119899 and 1199091 119909

2 119909

119898isin R

(1199091+ sdot sdot sdot + 119909

119899)119898

= sum

1198961+sdotsdotsdot+119896

119899=119898

(119898

1198961 119896

119899

) prod

1le119894le119899

119909119896119894

119894 (46)

where (119898

1198961 119896

119899) = 119898(119896

1 sdot sdot sdot 119896

119899) and 119896

1 119896

119899isin

0 1 119898

In particular

(1199091+ sdot sdot sdot + 119909

119899)4

le 4 sdot sum

1198961+sdotsdotsdot+119896

119899=4

1199091198961

1sdot sdot sdot 119909

119896119899

119899 (47)

which implies for 119878119899= sum

119899

119894=1119883

119894 that

1198641198784

119899le 4 sdot sum

1198961+sdotsdotsdot+119896

119899=4

119864 (1198831198961

1sdot sdot sdot 119883

119896119899

119899)

le 4 sdot

119899

sum

119905=1

sum

0le119894119895119896le119899minus1

10038161003816100381610038161003816119864 (119883

119905119883

119905+119894119883

119905+119894+119895119883

119905+119894+119895+119896)10038161003816100381610038161003816

(48)

Let us now introduce the following notation due to Doukhanand Louhichi (see [14]) for a sequence of centered rvrsquos119883

119905119899

119899ge1

119862119903119902

= sup100381610038161003816100381610038161003816Cov (119883

1199051

sdot sdot sdot 119883119905119898

119883119905119898+119903sdot sdot sdot 119883

119905119902

)100381610038161003816100381610038161003816 (49)

where the supremum runs over all divisions of the groupcomposed of 119902 rvrsquos into two subgroups such that the distancebetween the highest index of the rvrsquos in the first group and thelowest index of the rvrsquos from the second group is equal to 119903119903 isin 1 2 For 119903 = 0 we shall define 119862

0119902= 1 Let us then

put

1198831199051

= 119868 [1198801199051

le 119909] minus 119909 1198831199052

= 119868 [1198801199052

le 119909] minus 119909 (50)

We will now estimate the summands in (48)If max119894 119895 119896 = 119894 then

10038161003816100381610038161003816119864 119883

119905sdot (119883

119905+119894119883

119905+119894+119895119883

119905+119894+119895+119896)10038161003816100381610038161003816

=100381610038161003816100381610038161003816Cov (119883

119905 119883

119905+119894119883

119905+119894+119895119883

119905+119894+119895+119896)100381610038161003816100381610038161003816le 119862

1198944

(51)

Similarly for max119894 119895 119896 = 119896 we have

10038161003816100381610038161003816119864 (119883

119905119883

119905+119894119883

119905+119894+119895) sdot 119883

119905+119894+119895+11989610038161003816100381610038161003816

=10038161003816100381610038161003816Cov (119883

119905119883

119905+119894119883

119905+119894+119895 119883

119905+119894+119895+119896)10038161003816100381610038161003816le 119862

1198964

(52)

When max119894 119895 119896 = 119895 then

10038161003816100381610038161003816119864 (119883

119905119883

119905+119894) sdot (119883

119905+119894+119895119883

119905+119894+119895+119896)10038161003816100381610038161003816

=10038161003816100381610038161003816Cov (119883

119905119883

119905+119894 119883

119905+119894+119895119883

119905+119894+119895+119896)

+119864 (119883119905119883

119905+119894) 119864 (119883

119905+119894+119895119883

119905+119894+119895+119896)10038161003816100381610038161003816

le 1198621198954

+ 1198621198942119862

1198962

(53)

We keep on estimating the fourth moment of 119878119899by intro-

ducing 119905 isin 0 119899mdashthe index for which |119864(119883119905119883

119905+119894119883

119905+119894+119895

119883119905+119894+119895+119896

)| attains its maximum

1198641198784

119899le 4119899 sum

0le119894119895119896le119899minus1

10038161003816100381610038161003816119864 (119883

119905119883

119905+119894119883

119905+119894+119895119883

119905+119894+119895+119896)10038161003816100381610038161003816

=4119899( sum

0le119894119896le119895

[1198621198942sdot 119862

1198962+ 119862

1198954]+ sum

0le119894119895le119896

1198621198964+ sum

0le119896119895le119894

1198621198944)

= 4119899( sum

0le119894119896le119895

1198621198942sdot 119862

1198962+ 3 sum

0le119896119895le119894

1198621198944)

(54)

6 ISRN Probability and Statistics

The terms obtained in (54) may further be bounded fromabove in the following way

sum

0le119894119896le119895

1198621198942sdot 119862

1198962=

119899minus1

sum

119895=0

119895

sum

119894119896=0

1198621198942119862

1198962

le 119899

119899minus1

sum

119894119896=0

1198621198942119862

1198962= 119899(

119899minus1

sum

119894=0

1198621198942)

2

sum

0le119896119895le119894

1198621198944

=

119899minus1

sum

119894=0

119894

sum

119895119896=0

1198621198944

le

119899minus1

sum

119894=0

(119894 + 1) (119894 + 1) 1198621198944

=

119899minus1

sum

119894=0

(119894 + 1)2119862

1198944

(55)

We thus get the inequality

1198641198784

119899le 4([119899

119899minus1

sum

119894=0

1198621198942]

2

+ 3119899

119899minus1

sum

119894=0

(119894 + 1)2119862

1198944) (56)

We shall now focus on estimating 1198621199032

and 1198621199034 Since 119880

1199051

and 1198801199052

are associated uniformly [0 1] distributed rvrsquos 1198831199051

and 1198831199052

mdashas monotone functions of these rvrsquosmdashare associat-ed as well In order to bound

1198621199032

= sup0le1199051lt1199052le119899 1199052minus1199051=119903

Cov (1198831199051

1198831199052

) (57)

let us notice that from Schwarz inequality

Cov (1198831199051

1198831199052

) = 119864 (119868 [1198801199051

le 119909] 119868 [1198801199052

le 119909]) minus 1199092

le radic1199092 minus 1199092le 119909

(58)

On the other hand invoking inequalities from [13 15]

Cov (1198831199051

1198831199052

)

le sup119909isin[01]

[119875 (1198801199051

le 1199091198801199052

le 119909) minus 119875 (1198801199051

le 119909)119875 (1198801199052

le 119909)]

le 119862 sdot Cov13

(1198801199051

1198801199052

) for associated rvrsquos4 sdot Cov (119880

1199051

1198801199052

) for MTP2rvrsquos

(59)

As a consequence

1198621199032

le min 119909 119862 sdot Cov13

(1198801199051

1198801199052

) for associated rvrsquosmin 119909 4 sdot Cov (119880

1199051

1198801199052

) for MTP2rvrsquos

(60)

where 1199052minus 119905

1= 119903 Still we need the upper bound for 119862

1199034 It

turns out that119862

1199034

= sup1003816100381610038161003816100381610038161003816100381610038161003816

Cov2

prod

119894=1

(119868 [119880119905119894

le 119909] minus 119909)

4

prod

119894=3

(119868 [119880119905119894

le 119909] minus 119909)

1003816100381610038161003816100381610038161003816100381610038161003816

(61)

In other words among all divisions of the group 119883119905119894

119894 isin

1 2 3 4 into two subgroups 1198621199034

attains the biggest valuein case we take two subgroups consisted of two rvrsquos Thesupremum runs over the set 119905

1 119905

2 119905

3 119905

4isin N 0 le 119905

1lt 119905

2lt

1199053lt 119905

4le 119899 and 119905

3minus 119905

2= 119903 Elementary calculation leads to the

following formula

1198621199034

= sup 10038161003816100381610038161003816119875 (119880119905119894

le 119909 119894 isin 1 2 3 4)

minus 119875 (119880119905119894

le 119909 119894 isin 1 2) 119875 (119880119905119894

le 119909 119894 isin 3 4)

minus 119909 [119875 (119880119905119894

le 119909 119894 isin 1 2 3)

minus 119875 (119880119905119894

le 119909 119894 isin 1 2) 119875 (1198801199053

le 119909)

+ 119875 (119880119905119894

le 119909 119894 isin 1 3 4)

minus 119875 (119880119905119894

le 119909 119894 isin 3 4) 119875 (1198801199051

le 119909)

+ 119875 (119880119905119894

le 119909 119894 isin 1 2 4)

minus 119875 (119880119905119894

le 119909 119894 isin 1 2) 119875 (1198801199054

le 119909)

+ 119875 (119880119905119894

le 119909 119894 isin 2 3 4)

minus119875 (119880119905119894

le 119909 119894 isin 3 4) 119875 (1198801199052

le 119909)]

+ 1199092[119875 (119880

1199051

le 1199091198801199053

le 119909)

minus 119875 (1198801199051

le 119909)119875 (1198801199053

le 119909)

+ 119875 (1198801199051

le 1199091198801199054

le 119909)

minus 119875 (1198801199051

le 119909)119875 (1198801199054

le 119909)

+ 119875 (1198801199052

le 1199091198801199053

le 119909)

minus 119875 (1198801199052

le 119909)119875 (1198801199053

le 119909)

+ 119875 (1198801199052

le 1199091198801199054

le 119909)

minus119875 (1198801199052

le 119909)119875 (1198801199052

le 119909)]10038161003816100381610038161003816

le10038161003816100381610038161198771

1003816100381610038161003816 +10038161003816100381610038161198772

1003816100381610038161003816 +10038161003816100381610038161198773

1003816100381610038161003816

(62)

where for the sake of simplicity 1198771 119877

2 and 119877

3are respec-

tively the free coefficient the expression with 119909 and theexpression with 119909

2 Using the Lebowitz inequality (see [15]for instance) and inequalities obtained in [13 15] we arrive at10038161003816100381610038161198771

1003816100381610038161003816 =10038161003816100381610038161003816119875 (119880

119905119894

le 119909 119894 isin 1 2 3 4)

minus119875 (119880119905119894

le 119909 119894 isin 1 2) 119875 (119880119905119894

le 119909 119894 isin 3 4)10038161003816100381610038161003816

le 11986711988011990511198801199053

+ 11986711988011990511198801199054

+ 11986711988011990521198801199053

+ 11986711988011990521198801199054

ISRN Probability and Statistics 7

le

119862(

2

sum

119894=1

4

sum

119895=3

Cov13(119880

119905119894

119880119905119895

)) for associated rvrsquos

4(

2

sum

119894=1

4

sum

119895=3

Cov (119880119905119894

119880119905119895

)) for MTP2rvrsquos

(63)

Analogously we get

10038161003816100381610038161198772

1003816100381610038161003816

le

2119909 sdot 119862(

2

sum

119894=1

4

sum

119895=3

Cov13(119880

119905119894

119880119905119895

)) for associated rvrsquos

2119909 sdot 4(

2

sum

119894=1

4

sum

119895=3

Cov (119880119905119894

119880119905119895

)) for MTP2rvrsquos

10038161003816100381610038161198773

1003816100381610038161003816

le

1199092sdot 119862(

2

sum

119894=1

4

sum

119895=3

Cov13(119880

119905119894

119880119905119895

) ) for associated rvrsquos

1199092sdot 4(

2

sum

119894=1

4

sum

119895=3

Cov (119880119905119894

119880119905119895

)) for MTP2rvrsquos

(64)

Eventually we have

1198621199034

le

119891 (119909) sdot119862(

2

sum

119894=1

4

sum

119895=3

Cov13(119880

119905119894

119880119905119895

)) for associated rvrsquos

119891 (119909) sdot4(

2

sum

119894=1

4

sum

119895=3

Cov (119880119905119894

119880119905119895

)) for MTP2rvrsquos

(65)

where 1199053minus 119905

2= 119903 and 119891(119909) = (1 + 2119909 + 119909

2) for 119909 isin [0 1]

Let us now introduce the following notation

120579119903=

sup119905isinN

Cov (119880119905 119880

119905+119903) for 119903 isin 1 2

1 for 119903 = 0

(66)

and assume it decays powerly at rate 119886 in the following way

120579119903= 119874(

1

(119903 + 1)119886) for 119886 gt 0 119903 isin 0 1 (67)

Let us get back to inequality (56) we can now carry on Atfirst for associated rvrsquos

1198641198784

119899le 4[119899

119899minus1

sum

119903=0

min 119909 119862 sdot Cov13(119880

1199051

1198801199052

)]

2

+ 4 sdot 3119899

119899minus1

sum

119903=0

(119903 + 1)2119891 (119909) sdot 119862(

2

sum

119894=1

4

sum

119895=3

Cov13(119880

119905119894

119880119905119895

))

le 119863 sdot ([119899

119899minus1

sum

119903=0

min 119909 (119903 + 1)minus1198863

]

2

+119899

119899minus1

sum

119903=0

(119903 + 1)2(119903 + 1)

minus1198863)

= 119863 sdot ([119899

119899

sum

119903=1

min 119909 119903minus1198863]

2

+ 119899

119899

sum

119903=1

1199032119903minus1198863

)

le 119863 sdot ([

[

119899 sum

119903lt119909minus3119886

119909 + 119899 sum

119903ge119909minus3119886

1

1199031198863]

]

2

+ 119899

119899

sum

119903=1

1199032minus1198863

)

le 1198631sdot (119899

2119909

2(119886minus3)119886+ 120585)

(68)

where119863 and1198631are constants and

120585 = 119899

119899

sum

119903=1

1199032minus1198863

=

119874(119899) 119886 gt 9

119874 (119899 ln 119899) 119886 = 9

119874 (1198994minus1198863

) 3 lt 119886 lt 9

(69)

It is worth mentioning that in the last inequality of (68) weused the estimate

int

infin

119909

1

119905119901119889119905 sim

1

119909119901minus1for 119901 gt 1 (70)

At the same time in the case of MTP2rvrsquos we get

1198641198784

119899le 119863

2sdot (119899

2119909

2(119886minus1)119886+ 120577) (71)

where1198632is constant and

120577 = 119899

119899

sum

119903=1

1199032minus119886

=

119874(119899) 119886 gt 3

119874 (119899 ln 119899) 119886 = 3

119874 (1198994minus119886

) 1 lt 119886 lt 3

(72)

Let now 120572119899(119909) = (1radic119899)sum

119899

119894=1(119868[119880

119905119894

le 119909] minus 119909) Then

120572119899(119909) minus 120572

119899(119910) =

1

radic119899

119899

sum

119894=1

(119868 [119909 lt 119880119905119894

le 119910] minus (119909 minus 119910))

for 119909 119910 isin [0 1]

(73)

8 ISRN Probability and Statistics

has the fourth moment estimatedmdashin the case of associatedrvrsquosmdashby

119864[120572119899(119909) minus 120572

119899(119910)]

4

= 119864[1

radic119899

119899

sum

119894=1

(119868 [119909 lt 119880119905119894

le 119910] minus (119909 minus 119910))]

4

=1

1198992119864[

119899

sum

119894=1

(119868 [119909 lt 119880119905119894

le 119910] minus (119909 minus 119910))]

4

le 1198631sdot (

1

119899211989921003816100381610038161003816119909 minus 119910

1003816100381610038161003816

2(119886minus3)119886

+120585

1198992)

= 1198631sdot (

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

2(119886minus3)119886

+120585

1198992)

=

119874(119899minus1) 119886 gt 9

119874(ln 119899119899

) 119886 = 9

119874 (1198992minus1198863

) 3 lt 119886 lt 9

(74)

and in the case of MTP2rvrsquos by

119864[120572119899(119909) minus 120572

119899(119910)]

4

le 1198632sdot (

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

2(119886minus1)119886

+120577

1198992)

=

119874(119899minus1) 119886 gt 3

119874(ln 119899119899

) 119886 = 3

119874 (1198992minus119886

) 1 lt 119886 lt 3

(75)

In light of the Shao and Yursquos criterion our process is tight forassociated rvrsquos when 119886 gt 6 and for MTP

2rvrsquos when 119886 gt 2

Let us sum up this result in the following theorem

Theorem 3 Let 120572119899(119909) = (1radic119899)sum

119899

119894=1(119868[119880

119894le 119909] minus 119909) be the

empirical process built on an associated sequence of uniformly[0 1] distributed rvrsquos 119880

119894119894ge1

Let also

120579119903= sup

119905isinNCov (119880

119905 119880

119905+119903) = 119874 ((119903 + 1)

minus119886)

where 119903 isin 0 1 119886 gt 0

(76)

Then 120572119899(119909)

119899ge1is tight for 119886 gt 6 If the rvrsquos 119880

119894119894ge1

are MTP2

then the process is tight for 119886 gt 2

Yu assumed stationarity of 119880119894119894ge1

andsum

infin

119899=111989965+]Cov(119880

0 119880

119899) lt infin for a positive constant ]

thus the rate of decay 119886 gt 7 5 Our result weakens consid-erably these assumptions especially in the case of MTP

2rvrsquos

Louhichi in [16] proposed a different tightness criterioninvolving the so-called bracketing numbers She managedto enhance Yursquos resultmdasheven more than Shao and Yu in[8]mdashsince she proved that it suffices to take 119886 gt 4 to gettightness of the empirical process based on the associated

rvrsquos Nevertheless she kept the assumption of stationarityvalid

In the final analysis our resultrsquos advantage is the absenceof the stationarity assumption and the rate of decay for120579119903remains (up to the authorrsquos knowledge) unimproved for

MTP2rvrsquos

Unfortunately with a view to obtainingweak convergenceof the process in question that is also convergence offinite-dimensional distributions we do not know how tomanage without the assumption of stationarityTherefore weconclude with the following corollary

Corollary 4 Let 120572119899(119909) = (1radic119899)sum

119899

119894=1(119868[119880

119894le 119909] minus 119909) be the

empirical process built on a stationary associated sequence ofuniformly [0 1] distributed rvrsquos 119880

119894119894ge1

Let also

120579119903= Cov (119880

1 119880

1+119903) = 119874 ((119903 + 1)

minus119886)

where 119903 isin 0 1 119886 gt 0

(77)

Then if 119886 gt 6

120572119899(sdot) 997888rarr 119861 (sdot) weakly in 119863 [0 1] (78)

where 119861(sdot) is the zero mean Gaussian process on [0 1] withcovariance structure defined by

1205902(119909 119910) = 119909 and 119910 minus 119909119910

+

infin

sum

119895=1

[ Cov (119868 [1198801le 119909] 119868 [119880

119895+1le 119910])

+ Cov (119868 [1198801le 119910] 119868 [119880

119895+1le 119909])]

(79)

If the rvrsquos 119880119894119894ge1

are MTP2 then it suffices to be 119886 gt 2 in

order to claim the above convergence

Proof It remains to establish convergence of finite-dimen-sional distributions repeating the procedure from [7]

4 Tightness of the Kernel-TypeEmpirical Process

In this section we shall weaken assumption imposed on thecovariance structure of rvrsquos 119883

119895119895ge1

by Cai and Roussas in [1]for the kernel estimator of the df

119865119899(119909) =

1

119899

119899

sum

119895=1

119870(

119909 minus 119883119895

ℎ119899

) (80)

They deal with a stationary sequence of negatively associatedrvrsquos (cf [17]) and need the same condition as Yu [7] that is

1003816100381610038161003816Cov (1198831 119883

1+119899)1003816100381610038161003816 = 119874(

1

11989975+]) (81)

to get tightness of the smooth empirical process (see condi-tion (A4) in [1])

ISRN Probability and Statistics 9

It turns out that it suffices to have

1003816100381610038161003816Cov (1198801 119880

1+119899)1003816100381610038161003816 = 119874(

1

1198993119901(119901minus1)) (82)

where 119901 gt 4 is a positive constant taken from the tightnesscriterion (8) It is easy to see that asymptotically we get therate 3

On the way to prove it we will also take use of aRosenthal-type inequality due to Shao andYu (seeTheorem 2in [8]) we shall recall in the following lemma

Lemma 5 Let 119901 gt 2 and 119891 be a real valued function boundedby 1 with bounded first derivative Suppose that 119883

119899119899ge1

is asequence of stationary and associated rvrsquos such that for 119899 isin N

Cov (1198831 119883

119899) = 119874 (119899

minus119887) for some 119887 gt 119901 minus 1 (83)

Then for any 120583 gt 0 there exists some positive constant 119896120583

independent of the function 119891 for which

1198641003816100381610038161003816119878119899

(119891) minus 119864119878119899(119891)

1003816100381610038161003816

119901

le 119896120583(119899

1+12058310038171003817100381710038171003817119891

101584010038171003817100381710038171003817

2

+1198991199012 [

[

119899

sum

119895=1

10038161003816100381610038161003816Cov (119891 (119883

1)119891 (119883

119895))10038161003816100381610038161003816]

]

)

(84)

As we can see the lemma assumes association but itworks for negatively associated rvrsquos as well since in the proofit reaches back the result of Newman (see Proposition 15 in[18]) where both types of association are allowed

Let us recall that 120572119899(119909) = radic119899(119865

119899(119909) minus 119864119865

119899(119909)) where

119865119899(119909) = (1119899)sum

119899

119895=1119870((119909 minus 119883

119895)ℎ

119899) It is easy to see that

1198641003816100381610038161003816120572119899

(119909) minus 120572119899(119910)

1003816100381610038161003816

119901

= 119899minus1199012

11986410038161003816100381610038161003816119878119899(119891 (119883

119895)) minus 119864119878

119899(119891 (119883

119895))10038161003816100381610038161003816

119901

(85)

where

119891 (119883119895) = 119870(

119909 minus 119883119895

ℎ119899

) minus 119870(

119910 minus 119883119895

ℎ119899

)

119878119899(119891) =

119899

sum

119895=1

119891 (119883119895)

(86)

With an intent to use Lemma 5 we need 119891 to be boundedfrom above by 1 (which in our case is obvious) and to havebounded 119891

1015840 thus we assume

sup119905isinR

10038161003816100381610038161003816119870

1015840(119905)

10038161003816100381610038161003816

1

ℎ119899

le119862

119870

ℎ119899

lt infin (87)

Now applying inequality (84) we have

1198641003816100381610038161003816120572119899

(119909) minus 120572119899(119910)

1003816100381610038161003816

119901

le119896120583119899minus1199012

1198991+120583

1198622

119870

ℎ2

119899

+1198991199012[

[

119899

sum

119895=1

10038161003816100381610038161003816Cov (119891 (119883

1)119891 (119883

119895))10038161003816100381610038161003816]

]

1199012

(88)

Newman showed in [18] that

Cov (1198911(119883) 119891

2(119884))

= ∬R2119891

1015840

1(119909) 119891

1015840

2(119910)

times [119875 (119883 le 119909 119884 le 119910) minus 119875 (119883 le 119909) 119875 (119884 le 119910)] 119889119909 119889119910

(89)

if 1198911and 119891

2are real valued functions on R having square

integrable derivatives 1198911015840

1and 119891

1015840

2 respectively and provided

that 1198911(119883) 119891

2(119884) have finite secondmoments In light of that

equation and linearity of covariance we get10038161003816100381610038161003816Cov (119891 (119883

1) 119891 (119883

119895))10038161003816100381610038161003816

=

1003816100381610038161003816100381610038161003816∬

R2[119875 (119883

1lt 119909 minus ℎ

119899119904 119883

119895lt 119909 minus ℎ

119899119905)

minus119875 (1198831lt 119909 minus ℎ

119899119904 119883

119895lt 119910 minus ℎ

119899119905)]

minus [119875 (1198831lt 119909 minus ℎ

119899119904) 119875 (119883

119895lt 119909 minus ℎ

119899119905)

minus119875 (1198831lt 119909 minus ℎ

119899119904) 119875 (119883

119895lt 119910 minus ℎ

119899119905)]

minus [119875 (1198831lt 119910 minus ℎ

119899119904 119883

119895lt 119909 minus ℎ

119899119905)

minus119875 (1198831lt 119910 minus ℎ

119899119904 119883

119895lt 119910 minus ℎ

119899119905)]

minus [119875 (1198831lt 119910 minus ℎ

119899119904) 119875 (119883

119895lt 119910 minus ℎ

119899119905)

minus 119875 (1198831lt 119910 minus ℎ

119899119904)

times 119875 (119883119895lt 119909 minus ℎ

119899119905)] 119896 (119904) 119896 (119905) 119889119904 119889119905

10038161003816100381610038161003816100381610038161003816

(90)

Without the loss of generality we may and do assume that119909 gt 119910 thus10038161003816100381610038161003816Cov (119891 (119883

1) 119891 (119883

119895))10038161003816100381610038161003816

=

1003816100381610038161003816100381610038161003816∬

R2119875 (119910 minus ℎ

119899119904 le 119883

1lt 119909 minus ℎ

119899119904

119910 minus ℎ119899119905 le 119883

119895lt 119909 minus ℎ

119899119905)

minus 119875 (119910 minus ℎ119899119904 le 119883

1lt 119909 minus ℎ

119899119904)

times 119875 (119910 minus ℎ119899119905 le 119883

119895lt 119909 minus ℎ

119899119905) 119896 (119904) 119896 (119905) 119889119904 119889119905

1003816100381610038161003816100381610038161003816

(91)

and by triangle inequality we get10038161003816100381610038161003816Cov (119891 (119883

1) 119891 (119883

119895))10038161003816100381610038161003816

le ∬R2[int

119909minusℎ119899119904

119910minusℎ119899119904

int

119909minusℎ119899119905

119910minusℎ119899119905

1198911198831119883119895(119906 V) 119889119906 119889V

+int

119909minusℎ119899119904

119910minusℎ119899119904

1198911198831(119906) 119889119906int

119909minusℎ119899119905

119910minusℎ119899119905

119891119883119895(V) 119889V] 119896 (119904) 119896 (119905) 119889119904 119889119905

(92)

10 ISRN Probability and Statistics

1198911198831119883119895

(119906 V) 1198911198831

(119906) and 119891119883119895

(V) are the joint pdf of [1198831 119883

119895]

and marginal pdfrsquos of rvrsquos1198831and119883

119895 respectively We need

to further assume that 1198621stands for a common upper bound

of 1198911198831

and 1198911198831119883119895

for all 119895 isin N that is

100381710038171003817100381710038171198911198831

10038171003817100381710038171003817le 119862

1

1003817100381710038171003817100381710038171198911198831119883119895

100381710038171003817100381710038171003817le 119862

1forall119895 ge 1 (93)

Then we finally obtain

10038161003816100381610038161003816Cov (119891 (119883

1) 119891 (119883

119895))10038161003816100381610038161003816le 2119862

1

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

2

where = max 1198621 119862

2

1

(94)

On the other hand proceeding like Cai and Roussas in [1] wecan shortly get

10038161003816100381610038161003816Cov (119891 (119883

1) 119891 (119883

119895))10038161003816100381610038161003816

le

100381610038161003816100381610038161003816100381610038161003816

Cov(119870(119909 minus 119883

1

ℎ119899

) 119870(

119909 minus 119883119895

ℎ119899

))

100381610038161003816100381610038161003816100381610038161003816

+

100381610038161003816100381610038161003816100381610038161003816

Cov(119870(119909 minus 119883

1

ℎ119899

) 119870(

119910 minus 119883119895

ℎ119899

))

100381610038161003816100381610038161003816100381610038161003816

+

100381610038161003816100381610038161003816100381610038161003816

Cov(119870(119910 minus 119883

1

ℎ119899

) 119870(

119909 minus 119883119895

ℎ119899

))

100381610038161003816100381610038161003816100381610038161003816

+

100381610038161003816100381610038161003816100381610038161003816

Cov(119870(119910 minus 119883

1

ℎ119899

) 119870(

119910 minus 119883119895

ℎ119899

))

100381610038161003816100381610038161003816100381610038161003816

le 41198622

10038161003816100381610038161003816Cov (119883

1 119883

119895)10038161003816100381610038161003816

13

(95)

where 1198622is a constant relevant to the covariance inequality

for negatively associated rvrsquos (see [13])We now arrive at the following inequality

10038161003816100381610038161003816Cov (119891 (119883

1) 119891 (119883

119895))10038161003816100381610038161003816

le max 1198621 119862

2

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

2

10038161003816100381610038161003816Cov(119883

1 119883

119895)10038161003816100381610038161003816

13

(96)

Assuming |Cov(119891(1198831) 119891(119883

119895))| = 119874(119895

minus119903) and proceeding

similarly to (68) we can get

[

[

119899

sum

119895=1

10038161003816100381610038161003816Cov (119891 (119883

1) 119891 (119883

119895))10038161003816100381610038161003816]

]

1199012

le max 1198621 119862

2 [

[

sum

119895lt|119909minus119910|minus2(1199033)

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

2

+ sum

119895ge|119909minus119910|minus2(1199033)

1

1198951199033]

]

1199012

le 2max 1198621 119862

21003816100381610038161003816119909 minus 119910

1003816100381610038161003816

((119903minus3)119903)119901

(97)

To sum up we obtain the following inequality

1198641003816100381610038161003816120572119899

(119909) minus 120572119899(119910)

1003816100381610038161003816

119901

le 119896120583119899

1+120583minus1199012119862

2

119870

ℎ2

119899

+ 2max 1198621 119862

21003816100381610038161003816119909 minus 119910

1003816100381610038161003816

((119903minus3)119903)119901

le 1198621198991+120583minus1199012 1

ℎ2

119899

+1003816100381610038161003816119909 minus 119910

1003816100381610038161003816

((119903minus3)119903)119901

(98)

where 119862 = 119896120583max1198622

119896 2max119862

1 119862

2 The formula of the

tightness criterion (8) implies that ((119903 minus 3)119903)119901 gt 1 so

119903 gt3119901

119901 minus 1 where 119901 gt 2 (99)

Let us recall the assumptions imposed on the bandwidhtsℎ

119899119899ge1

by Cai and Roussas in [1]B1 lim

119899rarrinfinℎ119899= 0 and ℎ

119899gt 0 for all 119899 isin N

+

B2 lim119899rarrinfin

119899ℎ119899= infin thus ℎ

119899= 119874(1119899

1minus120573) 120573 gt 0

B3 lim119899rarrinfin

119899ℎ4

119899= 0 hence ℎ

119899= 119874(1119899

14+120575) 120575 gt 0

In light of the above

1198991+120583minus1199012 1

ℎ2

119899

= 11989932+120583+2120575minus1199012

(100)

where 120583 gt 0 119901 gt 2 and 0 lt 120575 lt 34 Confrontation with thetightness criterion (8) forces

3

2+ 120583 + 2120575 minus

119901

2lt minus

1

2 (101)

which implies 119901 gt 4 Let us conclude with the followingtheorem

Theorem6 Let 120572119899(119909) = radic119899(119865

119899(119909)minus119864119865

119899(119909)) be the empirical

process built on the kernel estimator of the df119865 for a stationarysequence of negatively associated rvrsquos Assume that conditionsA1 A2 (87) B1 B2 B3 are satisfied Then provided that thetightness criterion (8) holds with 119901 gt 4 it suffices to demand

1003816100381610038161003816Cov (1198801 119880

1+119899)1003816100381610038161003816 = 119874(

1

1198993119901(119901minus1)) (102)

in order to obtain120572

119899(sdot) 997888rarr 119861 (sdot) weakly in 119863 [0 1] (103)

where 119861(sdot) is the zero mean Gaussian process with covariancestructure defined by

1205902(119909 119910) = 119865 (119909 and 119910) minus 119865 (119909) 119865 (119910)

+

infin

sum

119895=1

[ Cov (119868 [1198831le 119909] 119868 [119883

119895+1le 119910])

+ Cov (119868 [1198831le 119910] 119868 [119883

119895+1le 119909])]

(104)

and 119909 119910 isin [0 1]

Proof The remaining covergence of finite-dimensional dis-tributions of 120572

119899(119909) is established in [1]

ISRN Probability and Statistics 11

5 Weak Convergence of theRecursive Kernel-Type Empirical Processunder IID Assumption

The aim of this section is to show weak convergence of theempirical process

120572119899(119909) =

1

radic119899

119899

sum

119895=1

[119870(

119909 minus 119883119895

ℎ119895

) minus 119864119870(

119909 minus 119883119895

ℎ119895

)] (105)

built on iid rvrsquos 119883119895119895ge1

Wewill prove that120572

119899(sdot) convergesweakly to the Brownian

bridge 119861(sdot) with the following covariance structure

1205902(119909 119910) = 119865 (119909 and 119910) minus 119865 (119909) 119865 (119910) for 119909 119910 isin [0 1]

(106)

It turns out that the tightness criterion (8) suffices to reachthe goal Convergence of finite-dimensional distributions of120572

119899(sdot) holds and we shall show itProceeding like Cai and Roussas in [1] we need to show

that for any 119886 119887 isin R

119886120572119899(119909) + 119887120572

119899(119910) 997888rarr 119886119861 (119909) + 119887119861 (119910) in distribution

(107)

Let us introduce the notation

119884119894(119909) = 119870(

119909 minus 119883119894

ℎ119894

) minus 119864119870(119909 minus 119883

119894

ℎ119894

) (108)

and look into the covariance structure of 120572119899(sdot)

Cov (120572119899(119909) 120572

119899(119910))

= Cov( 1

radic119899

119899

sum

119894=1

119884119894(119909)

1

radic119899

119899

sum

119894=1

119884119894(119910))

=1

119899

119899

sum

119894=1

Cov (119884119894(119909) 119884

119894(119910))

=1

119899

119899

sum

119894=1

Cov(119870(119909 minus 119883

119894

ℎ119894

) 119870(119910 minus 119883

119894

ℎ119894

))

(109)

where the second equality follows from assumed indepen-dence of rvrsquos 119883

119894119894ge1

Firstly we observe that each summand converges to119865(119909and

119910) minus 119865(119909)119865(119910) since

Cov(119870(119909 minus 119883

119894

ℎ119894

) 119870(119910 minus 119883

119894

ℎ119894

))

= 119864[119870(119909 minus 119883

119894

ℎ119894

)119870(119910 minus 119883

119894

ℎ119894

)]

minus 119864119870(119909 minus 119883

119894

ℎ119894

)119864119870(119910 minus 119883

119894

ℎ119894

)

(110)

where

119864[119870(119909 minus 119883

119894

ℎ119894

)119870(119910 minus 119883

119894

ℎ119894

)]

= int

119909and119910

minusinfin

119870(119909 minus 119905

ℎ119894

)119870(119910 minus 119905

ℎ119894

)119889119865 (119905)

+ int

119909or119910

119909and119910

119870(119909 minus 119905

ℎ119894

)119870(119910 minus 119905

ℎ119894

)119889119865 (119905)

+ int

infin

119909or119910

119870(119909 minus 119905

ℎ119894

)119870(119910 minus 119905

ℎ119894

)119889119865 (119905)

(111)

and 119865 denotes the common df of the rvrsquos 119883119894119894ge1

In [6] itwas shown that

119864119870(119909 minus 119883

119894

ℎ119894

) 997888rarr 119865 (119909) (112)

and recalling that the kernel function 119870 is a df as well wearrive at the conclusion

Secondly applying Toeplitz lemma we get

Cov (120572119899(119909) 120572

119899(119910)) 997888rarr 119865 (119909 and 119910) minus 119865 (119909) 119865 (119910)

= 1205902(119909 119910) 119899 997888rarr infin

(113)

Thus

Var (119886120572119899(119909) + 119887120572

119899(119910))

= 1198862 Var (120572

119899(119909)) + 2119886119887Cov (120572

119899(119909) 120572

119899(119910))

+ 1198872 Var (120572

119899(119910))

997888rarr 1198862120590

2(119909 119909) + 2119886119887120590

2(119909 119910) + 119887

2120590

2(119910 119910) 119899 997888rarr infin

(114)

We are now on the way to prove that 119886120572119899(119909) + 119887120572

119899(119910) con-

verges in distribution to 119886119861(119909) + 119887119861(119910) sim N(0 1198862120590

2(119909 119909) +

21198861198871205902(119909 119910) + 119887

2120590

2(119910 119910)) which in other words means that

1

radic119899

infin

sum

119894=1

119886119884119894(119909) + 119887119884

119894(119910)

radicVar (119886120572119899(119909) + 119887120572

119899(119910))

997888rarr N (0 1) (115)

In order to obtain (115) we will use Lyapunov condition

lim119899rarrinfin

1

1198632+120575

119899

119899

sum

119894=1

1198641003816100381610038161003816119886119884119894

(119909) + 119887119884119894(119910)

1003816100381610038161003816

2+120575

= 0 (116)

for 120575 = 1 where 1198632

119899= Var(sum119899

119894=1[119886119884

119894(119909) + 119887119884

119894(119910)]) Since

|119870(sdot) minus 119864119870(sdot)| le 2

1198641003816100381610038161003816119886119884119894

(119909) + 119887119884i (119910)1003816100381610038161003816

3

le 8 (1198863+ 119887

3) (117)

Moreover 1198632

119899= 119899Var(119886120572

119899(119909) + 119887120572

119899(119910)) together with (114)

yields

1198633

119899= 119874 (119899

32) (118)

12 ISRN Probability and Statistics

Combining (117) with (118) we obtain

1

1198633

119899

119899

sum

119894=1

1198641003816100381610038161003816119886119884119894

(119909) + 119887119884119894(119910)

1003816100381610038161003816

3

= 119874 (119899minus12

) (119)

which shows that Lyapunov condition holds and completesthe proof of convergence of finite-dimensional distributionsof 120572

119899(119909)We summarize the result of that section in the following

theorem

Theorem 7 Let 120572119899(119909) be the recursive kernel-type empirical

process defined by (105) built on iid rvrsquos 119883119895119895ge1

withmarginal df 119865 If 120572

119899(sdot) satisfies the tightness criterion (8) then

120572119899(sdot) 997888rarr 119861 (sdot) weakly in 119863 [0 1] (120)

where 119861 is the Brownian bridge

Acknowledgment

The author would like to thank the referees for the carefulreading of the paper and helpful remarks

References

[1] Z Cai and G G Roussas ldquoWeak convergence for smoothestimator of a distribution function under negative associationrdquoStochastic Analysis and Applications vol 17 no 2 pp 145ndash1681999

[2] Z Cai andG G Roussas ldquoEfficient Estimation of a DistributionFunction under Qudrant Dependencerdquo Scandinavian Journal ofStatistics vol 25 no 1 pp 211ndash224 1998

[3] ZW Cai and G G Roussas ldquoUniform strong estimation under120572-mixing with ratesrdquo Statistics amp Probability Letters vol 15 no1 pp 47ndash55 1992

[4] G G Roussas ldquoKernel estimates under association stronguniform consistencyrdquo Statistics amp Probability Letters vol 12 no5 pp 393ndash403 1991

[5] Y F Li and S C Yang ldquoUniformly asymptotic normalityof the smooth estimation of the distribution function underassociated samplesrdquo Acta Mathematicae Applicatae Sinica vol28 no 4 pp 639ndash651 2005

[6] Y Li C Wei and S Yang ldquoThe recursive kernel distributionfunction estimator based onnegatively and positively associatedsequencesrdquo Communications in Statistics vol 39 no 20 pp3585ndash3595 2010

[7] H Yu ldquoA Glivenko-Cantelli lemma and weak convergence forempirical processes of associated sequencesrdquo ProbabilityTheoryand Related Fields vol 95 no 3 pp 357ndash370 1993

[8] Q M Shao and H Yu ldquoWeak convergence for weightedempirical processes of dependent sequencesrdquo The Annals ofProbability vol 24 no 4 pp 2098ndash2127 1996

[9] P Billingsley Convergence of Probability Measures Wiley Seriesin Probability and Statistics Probability and Statistics JohnWiley amp Sons New York NY USA 1999

[10] J D Esary F Proschan and D W Walkup ldquoAssociation ofrandom variables with applicationsrdquo Annals of MathematicalStatistics vol 38 pp 1466ndash1474 1967

[11] A Bulinski and A Shashkin Limit Theorems for AssociatedRandom Fields and Related Systems vol 10 of Advanced Serieson Statistical Science and Applied Probability World Scientific2007

[12] S Karlin and Y Rinott ldquoClasses of orderings of measures andrelated correlation inequalities I Multivariate totally positivedistributionsrdquo Journal of Multivariate Analysis vol 10 no 4 pp467ndash498 1980

[13] P Matuła and M Ziemba ldquoGeneralized covariance inequali-tiesrdquo Central European Journal of Mathematics vol 9 no 2 pp281ndash293 2011

[14] P Doukhan and S Louhichi ldquoA new weak dependence condi-tion and applications to moment inequalitiesrdquo Stochastic Proc-esses and their Applications vol 84 no 2 pp 313ndash342 1999

[15] P Matuła ldquoA note on some inequalities for certain classes ofpositively dependent random variablesrdquo Probability and Math-ematical Statistics vol 24 no 1 pp 17ndash26 2004

[16] S Louhichi ldquoWeak convergence for empirical processes of asso-ciated sequencesrdquo Annales de lrsquoInstitut Henri Poincare vol 36no 5 pp 547ndash567 2000

[17] K Joag-Dev and F Proschan ldquoNegative association of randomvariables with applicationsrdquoThe Annals of Statistics vol 11 no1 pp 286ndash295 1983

[18] C M Newman ldquoAsymptotic independence and limit theoremsfor positively and negatively dependent random variablesrdquo inInequalities in Statistics and Probability Y L Tong Ed vol 5pp 127ndash140 Institute ofMathematical StatisticsHayward CalifUSA 1984

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

ISRN Probability and Statistics 5

= 119875( max1le119894le2(119898

119899+1)

1003816100381610038161003816120572119899(119910 + 119894119903

119899) minus 120572

119899(119910)

1003816100381610038161003816 ge 119872120598)

le

119862119901min119901

11199012+1199013

(119872120598)119901

(2120575)min119901

11199012+1199013le 120578

(43)

for any fixed 119910 isin [0 1] Since the upper bound does notdepend on 119910 and the probability measure as well as supre-mum function are continuous we obtain

119875( sup0le119910le1minus2120575

sup119910le119909le119910+2120575

1003816100381610038161003816120572119899(119909) minus 120572

119899(119910)

1003816100381610038161003816 ge 4119872120598 + 119862radic119899ℎ4

119899) le 120578

(44)

where 4119872120598 + 119862radic119899ℎ4

119899is arbitrarily small

Since condition (10) is checked the proof is completed

3 Tightness of the Standard Empirical Process

In this section we deal with the standard empirical processbuilt on an associated sequence of uniformly [0 1]distributedrvrsquos 119880

119895119895ge1

that is

120572119899(119909) =

1

radic119899

119899

sum

119895=1

(119868 [119880119895le 119909] minus 119909) (45)

where 119909 isin [0 1] We shall relax the restrictions imposedon the process by Yu in [7] to obtain tightness Preciselywe do not need stationarity any more due to the techniquedrawn from [14] and we lower the assumed rate at which thecovariance tends to zero

While proving tightness of the empirical process we willuse the criterion proved in the first section as well as some ofour covariance inequalities

We shall start with the fact known under the name ofmultinomial theorem We recall it in the following lemma

Lemma 2 For natural numbers119898 119899 and 1199091 119909

2 119909

119898isin R

(1199091+ sdot sdot sdot + 119909

119899)119898

= sum

1198961+sdotsdotsdot+119896

119899=119898

(119898

1198961 119896

119899

) prod

1le119894le119899

119909119896119894

119894 (46)

where (119898

1198961 119896

119899) = 119898(119896

1 sdot sdot sdot 119896

119899) and 119896

1 119896

119899isin

0 1 119898

In particular

(1199091+ sdot sdot sdot + 119909

119899)4

le 4 sdot sum

1198961+sdotsdotsdot+119896

119899=4

1199091198961

1sdot sdot sdot 119909

119896119899

119899 (47)

which implies for 119878119899= sum

119899

119894=1119883

119894 that

1198641198784

119899le 4 sdot sum

1198961+sdotsdotsdot+119896

119899=4

119864 (1198831198961

1sdot sdot sdot 119883

119896119899

119899)

le 4 sdot

119899

sum

119905=1

sum

0le119894119895119896le119899minus1

10038161003816100381610038161003816119864 (119883

119905119883

119905+119894119883

119905+119894+119895119883

119905+119894+119895+119896)10038161003816100381610038161003816

(48)

Let us now introduce the following notation due to Doukhanand Louhichi (see [14]) for a sequence of centered rvrsquos119883

119905119899

119899ge1

119862119903119902

= sup100381610038161003816100381610038161003816Cov (119883

1199051

sdot sdot sdot 119883119905119898

119883119905119898+119903sdot sdot sdot 119883

119905119902

)100381610038161003816100381610038161003816 (49)

where the supremum runs over all divisions of the groupcomposed of 119902 rvrsquos into two subgroups such that the distancebetween the highest index of the rvrsquos in the first group and thelowest index of the rvrsquos from the second group is equal to 119903119903 isin 1 2 For 119903 = 0 we shall define 119862

0119902= 1 Let us then

put

1198831199051

= 119868 [1198801199051

le 119909] minus 119909 1198831199052

= 119868 [1198801199052

le 119909] minus 119909 (50)

We will now estimate the summands in (48)If max119894 119895 119896 = 119894 then

10038161003816100381610038161003816119864 119883

119905sdot (119883

119905+119894119883

119905+119894+119895119883

119905+119894+119895+119896)10038161003816100381610038161003816

=100381610038161003816100381610038161003816Cov (119883

119905 119883

119905+119894119883

119905+119894+119895119883

119905+119894+119895+119896)100381610038161003816100381610038161003816le 119862

1198944

(51)

Similarly for max119894 119895 119896 = 119896 we have

10038161003816100381610038161003816119864 (119883

119905119883

119905+119894119883

119905+119894+119895) sdot 119883

119905+119894+119895+11989610038161003816100381610038161003816

=10038161003816100381610038161003816Cov (119883

119905119883

119905+119894119883

119905+119894+119895 119883

119905+119894+119895+119896)10038161003816100381610038161003816le 119862

1198964

(52)

When max119894 119895 119896 = 119895 then

10038161003816100381610038161003816119864 (119883

119905119883

119905+119894) sdot (119883

119905+119894+119895119883

119905+119894+119895+119896)10038161003816100381610038161003816

=10038161003816100381610038161003816Cov (119883

119905119883

119905+119894 119883

119905+119894+119895119883

119905+119894+119895+119896)

+119864 (119883119905119883

119905+119894) 119864 (119883

119905+119894+119895119883

119905+119894+119895+119896)10038161003816100381610038161003816

le 1198621198954

+ 1198621198942119862

1198962

(53)

We keep on estimating the fourth moment of 119878119899by intro-

ducing 119905 isin 0 119899mdashthe index for which |119864(119883119905119883

119905+119894119883

119905+119894+119895

119883119905+119894+119895+119896

)| attains its maximum

1198641198784

119899le 4119899 sum

0le119894119895119896le119899minus1

10038161003816100381610038161003816119864 (119883

119905119883

119905+119894119883

119905+119894+119895119883

119905+119894+119895+119896)10038161003816100381610038161003816

=4119899( sum

0le119894119896le119895

[1198621198942sdot 119862

1198962+ 119862

1198954]+ sum

0le119894119895le119896

1198621198964+ sum

0le119896119895le119894

1198621198944)

= 4119899( sum

0le119894119896le119895

1198621198942sdot 119862

1198962+ 3 sum

0le119896119895le119894

1198621198944)

(54)

6 ISRN Probability and Statistics

The terms obtained in (54) may further be bounded fromabove in the following way

sum

0le119894119896le119895

1198621198942sdot 119862

1198962=

119899minus1

sum

119895=0

119895

sum

119894119896=0

1198621198942119862

1198962

le 119899

119899minus1

sum

119894119896=0

1198621198942119862

1198962= 119899(

119899minus1

sum

119894=0

1198621198942)

2

sum

0le119896119895le119894

1198621198944

=

119899minus1

sum

119894=0

119894

sum

119895119896=0

1198621198944

le

119899minus1

sum

119894=0

(119894 + 1) (119894 + 1) 1198621198944

=

119899minus1

sum

119894=0

(119894 + 1)2119862

1198944

(55)

We thus get the inequality

1198641198784

119899le 4([119899

119899minus1

sum

119894=0

1198621198942]

2

+ 3119899

119899minus1

sum

119894=0

(119894 + 1)2119862

1198944) (56)

We shall now focus on estimating 1198621199032

and 1198621199034 Since 119880

1199051

and 1198801199052

are associated uniformly [0 1] distributed rvrsquos 1198831199051

and 1198831199052

mdashas monotone functions of these rvrsquosmdashare associat-ed as well In order to bound

1198621199032

= sup0le1199051lt1199052le119899 1199052minus1199051=119903

Cov (1198831199051

1198831199052

) (57)

let us notice that from Schwarz inequality

Cov (1198831199051

1198831199052

) = 119864 (119868 [1198801199051

le 119909] 119868 [1198801199052

le 119909]) minus 1199092

le radic1199092 minus 1199092le 119909

(58)

On the other hand invoking inequalities from [13 15]

Cov (1198831199051

1198831199052

)

le sup119909isin[01]

[119875 (1198801199051

le 1199091198801199052

le 119909) minus 119875 (1198801199051

le 119909)119875 (1198801199052

le 119909)]

le 119862 sdot Cov13

(1198801199051

1198801199052

) for associated rvrsquos4 sdot Cov (119880

1199051

1198801199052

) for MTP2rvrsquos

(59)

As a consequence

1198621199032

le min 119909 119862 sdot Cov13

(1198801199051

1198801199052

) for associated rvrsquosmin 119909 4 sdot Cov (119880

1199051

1198801199052

) for MTP2rvrsquos

(60)

where 1199052minus 119905

1= 119903 Still we need the upper bound for 119862

1199034 It

turns out that119862

1199034

= sup1003816100381610038161003816100381610038161003816100381610038161003816

Cov2

prod

119894=1

(119868 [119880119905119894

le 119909] minus 119909)

4

prod

119894=3

(119868 [119880119905119894

le 119909] minus 119909)

1003816100381610038161003816100381610038161003816100381610038161003816

(61)

In other words among all divisions of the group 119883119905119894

119894 isin

1 2 3 4 into two subgroups 1198621199034

attains the biggest valuein case we take two subgroups consisted of two rvrsquos Thesupremum runs over the set 119905

1 119905

2 119905

3 119905

4isin N 0 le 119905

1lt 119905

2lt

1199053lt 119905

4le 119899 and 119905

3minus 119905

2= 119903 Elementary calculation leads to the

following formula

1198621199034

= sup 10038161003816100381610038161003816119875 (119880119905119894

le 119909 119894 isin 1 2 3 4)

minus 119875 (119880119905119894

le 119909 119894 isin 1 2) 119875 (119880119905119894

le 119909 119894 isin 3 4)

minus 119909 [119875 (119880119905119894

le 119909 119894 isin 1 2 3)

minus 119875 (119880119905119894

le 119909 119894 isin 1 2) 119875 (1198801199053

le 119909)

+ 119875 (119880119905119894

le 119909 119894 isin 1 3 4)

minus 119875 (119880119905119894

le 119909 119894 isin 3 4) 119875 (1198801199051

le 119909)

+ 119875 (119880119905119894

le 119909 119894 isin 1 2 4)

minus 119875 (119880119905119894

le 119909 119894 isin 1 2) 119875 (1198801199054

le 119909)

+ 119875 (119880119905119894

le 119909 119894 isin 2 3 4)

minus119875 (119880119905119894

le 119909 119894 isin 3 4) 119875 (1198801199052

le 119909)]

+ 1199092[119875 (119880

1199051

le 1199091198801199053

le 119909)

minus 119875 (1198801199051

le 119909)119875 (1198801199053

le 119909)

+ 119875 (1198801199051

le 1199091198801199054

le 119909)

minus 119875 (1198801199051

le 119909)119875 (1198801199054

le 119909)

+ 119875 (1198801199052

le 1199091198801199053

le 119909)

minus 119875 (1198801199052

le 119909)119875 (1198801199053

le 119909)

+ 119875 (1198801199052

le 1199091198801199054

le 119909)

minus119875 (1198801199052

le 119909)119875 (1198801199052

le 119909)]10038161003816100381610038161003816

le10038161003816100381610038161198771

1003816100381610038161003816 +10038161003816100381610038161198772

1003816100381610038161003816 +10038161003816100381610038161198773

1003816100381610038161003816

(62)

where for the sake of simplicity 1198771 119877

2 and 119877

3are respec-

tively the free coefficient the expression with 119909 and theexpression with 119909

2 Using the Lebowitz inequality (see [15]for instance) and inequalities obtained in [13 15] we arrive at10038161003816100381610038161198771

1003816100381610038161003816 =10038161003816100381610038161003816119875 (119880

119905119894

le 119909 119894 isin 1 2 3 4)

minus119875 (119880119905119894

le 119909 119894 isin 1 2) 119875 (119880119905119894

le 119909 119894 isin 3 4)10038161003816100381610038161003816

le 11986711988011990511198801199053

+ 11986711988011990511198801199054

+ 11986711988011990521198801199053

+ 11986711988011990521198801199054

ISRN Probability and Statistics 7

le

119862(

2

sum

119894=1

4

sum

119895=3

Cov13(119880

119905119894

119880119905119895

)) for associated rvrsquos

4(

2

sum

119894=1

4

sum

119895=3

Cov (119880119905119894

119880119905119895

)) for MTP2rvrsquos

(63)

Analogously we get

10038161003816100381610038161198772

1003816100381610038161003816

le

2119909 sdot 119862(

2

sum

119894=1

4

sum

119895=3

Cov13(119880

119905119894

119880119905119895

)) for associated rvrsquos

2119909 sdot 4(

2

sum

119894=1

4

sum

119895=3

Cov (119880119905119894

119880119905119895

)) for MTP2rvrsquos

10038161003816100381610038161198773

1003816100381610038161003816

le

1199092sdot 119862(

2

sum

119894=1

4

sum

119895=3

Cov13(119880

119905119894

119880119905119895

) ) for associated rvrsquos

1199092sdot 4(

2

sum

119894=1

4

sum

119895=3

Cov (119880119905119894

119880119905119895

)) for MTP2rvrsquos

(64)

Eventually we have

1198621199034

le

119891 (119909) sdot119862(

2

sum

119894=1

4

sum

119895=3

Cov13(119880

119905119894

119880119905119895

)) for associated rvrsquos

119891 (119909) sdot4(

2

sum

119894=1

4

sum

119895=3

Cov (119880119905119894

119880119905119895

)) for MTP2rvrsquos

(65)

where 1199053minus 119905

2= 119903 and 119891(119909) = (1 + 2119909 + 119909

2) for 119909 isin [0 1]

Let us now introduce the following notation

120579119903=

sup119905isinN

Cov (119880119905 119880

119905+119903) for 119903 isin 1 2

1 for 119903 = 0

(66)

and assume it decays powerly at rate 119886 in the following way

120579119903= 119874(

1

(119903 + 1)119886) for 119886 gt 0 119903 isin 0 1 (67)

Let us get back to inequality (56) we can now carry on Atfirst for associated rvrsquos

1198641198784

119899le 4[119899

119899minus1

sum

119903=0

min 119909 119862 sdot Cov13(119880

1199051

1198801199052

)]

2

+ 4 sdot 3119899

119899minus1

sum

119903=0

(119903 + 1)2119891 (119909) sdot 119862(

2

sum

119894=1

4

sum

119895=3

Cov13(119880

119905119894

119880119905119895

))

le 119863 sdot ([119899

119899minus1

sum

119903=0

min 119909 (119903 + 1)minus1198863

]

2

+119899

119899minus1

sum

119903=0

(119903 + 1)2(119903 + 1)

minus1198863)

= 119863 sdot ([119899

119899

sum

119903=1

min 119909 119903minus1198863]

2

+ 119899

119899

sum

119903=1

1199032119903minus1198863

)

le 119863 sdot ([

[

119899 sum

119903lt119909minus3119886

119909 + 119899 sum

119903ge119909minus3119886

1

1199031198863]

]

2

+ 119899

119899

sum

119903=1

1199032minus1198863

)

le 1198631sdot (119899

2119909

2(119886minus3)119886+ 120585)

(68)

where119863 and1198631are constants and

120585 = 119899

119899

sum

119903=1

1199032minus1198863

=

119874(119899) 119886 gt 9

119874 (119899 ln 119899) 119886 = 9

119874 (1198994minus1198863

) 3 lt 119886 lt 9

(69)

It is worth mentioning that in the last inequality of (68) weused the estimate

int

infin

119909

1

119905119901119889119905 sim

1

119909119901minus1for 119901 gt 1 (70)

At the same time in the case of MTP2rvrsquos we get

1198641198784

119899le 119863

2sdot (119899

2119909

2(119886minus1)119886+ 120577) (71)

where1198632is constant and

120577 = 119899

119899

sum

119903=1

1199032minus119886

=

119874(119899) 119886 gt 3

119874 (119899 ln 119899) 119886 = 3

119874 (1198994minus119886

) 1 lt 119886 lt 3

(72)

Let now 120572119899(119909) = (1radic119899)sum

119899

119894=1(119868[119880

119905119894

le 119909] minus 119909) Then

120572119899(119909) minus 120572

119899(119910) =

1

radic119899

119899

sum

119894=1

(119868 [119909 lt 119880119905119894

le 119910] minus (119909 minus 119910))

for 119909 119910 isin [0 1]

(73)

8 ISRN Probability and Statistics

has the fourth moment estimatedmdashin the case of associatedrvrsquosmdashby

119864[120572119899(119909) minus 120572

119899(119910)]

4

= 119864[1

radic119899

119899

sum

119894=1

(119868 [119909 lt 119880119905119894

le 119910] minus (119909 minus 119910))]

4

=1

1198992119864[

119899

sum

119894=1

(119868 [119909 lt 119880119905119894

le 119910] minus (119909 minus 119910))]

4

le 1198631sdot (

1

119899211989921003816100381610038161003816119909 minus 119910

1003816100381610038161003816

2(119886minus3)119886

+120585

1198992)

= 1198631sdot (

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

2(119886minus3)119886

+120585

1198992)

=

119874(119899minus1) 119886 gt 9

119874(ln 119899119899

) 119886 = 9

119874 (1198992minus1198863

) 3 lt 119886 lt 9

(74)

and in the case of MTP2rvrsquos by

119864[120572119899(119909) minus 120572

119899(119910)]

4

le 1198632sdot (

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

2(119886minus1)119886

+120577

1198992)

=

119874(119899minus1) 119886 gt 3

119874(ln 119899119899

) 119886 = 3

119874 (1198992minus119886

) 1 lt 119886 lt 3

(75)

In light of the Shao and Yursquos criterion our process is tight forassociated rvrsquos when 119886 gt 6 and for MTP

2rvrsquos when 119886 gt 2

Let us sum up this result in the following theorem

Theorem 3 Let 120572119899(119909) = (1radic119899)sum

119899

119894=1(119868[119880

119894le 119909] minus 119909) be the

empirical process built on an associated sequence of uniformly[0 1] distributed rvrsquos 119880

119894119894ge1

Let also

120579119903= sup

119905isinNCov (119880

119905 119880

119905+119903) = 119874 ((119903 + 1)

minus119886)

where 119903 isin 0 1 119886 gt 0

(76)

Then 120572119899(119909)

119899ge1is tight for 119886 gt 6 If the rvrsquos 119880

119894119894ge1

are MTP2

then the process is tight for 119886 gt 2

Yu assumed stationarity of 119880119894119894ge1

andsum

infin

119899=111989965+]Cov(119880

0 119880

119899) lt infin for a positive constant ]

thus the rate of decay 119886 gt 7 5 Our result weakens consid-erably these assumptions especially in the case of MTP

2rvrsquos

Louhichi in [16] proposed a different tightness criterioninvolving the so-called bracketing numbers She managedto enhance Yursquos resultmdasheven more than Shao and Yu in[8]mdashsince she proved that it suffices to take 119886 gt 4 to gettightness of the empirical process based on the associated

rvrsquos Nevertheless she kept the assumption of stationarityvalid

In the final analysis our resultrsquos advantage is the absenceof the stationarity assumption and the rate of decay for120579119903remains (up to the authorrsquos knowledge) unimproved for

MTP2rvrsquos

Unfortunately with a view to obtainingweak convergenceof the process in question that is also convergence offinite-dimensional distributions we do not know how tomanage without the assumption of stationarityTherefore weconclude with the following corollary

Corollary 4 Let 120572119899(119909) = (1radic119899)sum

119899

119894=1(119868[119880

119894le 119909] minus 119909) be the

empirical process built on a stationary associated sequence ofuniformly [0 1] distributed rvrsquos 119880

119894119894ge1

Let also

120579119903= Cov (119880

1 119880

1+119903) = 119874 ((119903 + 1)

minus119886)

where 119903 isin 0 1 119886 gt 0

(77)

Then if 119886 gt 6

120572119899(sdot) 997888rarr 119861 (sdot) weakly in 119863 [0 1] (78)

where 119861(sdot) is the zero mean Gaussian process on [0 1] withcovariance structure defined by

1205902(119909 119910) = 119909 and 119910 minus 119909119910

+

infin

sum

119895=1

[ Cov (119868 [1198801le 119909] 119868 [119880

119895+1le 119910])

+ Cov (119868 [1198801le 119910] 119868 [119880

119895+1le 119909])]

(79)

If the rvrsquos 119880119894119894ge1

are MTP2 then it suffices to be 119886 gt 2 in

order to claim the above convergence

Proof It remains to establish convergence of finite-dimen-sional distributions repeating the procedure from [7]

4 Tightness of the Kernel-TypeEmpirical Process

In this section we shall weaken assumption imposed on thecovariance structure of rvrsquos 119883

119895119895ge1

by Cai and Roussas in [1]for the kernel estimator of the df

119865119899(119909) =

1

119899

119899

sum

119895=1

119870(

119909 minus 119883119895

ℎ119899

) (80)

They deal with a stationary sequence of negatively associatedrvrsquos (cf [17]) and need the same condition as Yu [7] that is

1003816100381610038161003816Cov (1198831 119883

1+119899)1003816100381610038161003816 = 119874(

1

11989975+]) (81)

to get tightness of the smooth empirical process (see condi-tion (A4) in [1])

ISRN Probability and Statistics 9

It turns out that it suffices to have

1003816100381610038161003816Cov (1198801 119880

1+119899)1003816100381610038161003816 = 119874(

1

1198993119901(119901minus1)) (82)

where 119901 gt 4 is a positive constant taken from the tightnesscriterion (8) It is easy to see that asymptotically we get therate 3

On the way to prove it we will also take use of aRosenthal-type inequality due to Shao andYu (seeTheorem 2in [8]) we shall recall in the following lemma

Lemma 5 Let 119901 gt 2 and 119891 be a real valued function boundedby 1 with bounded first derivative Suppose that 119883

119899119899ge1

is asequence of stationary and associated rvrsquos such that for 119899 isin N

Cov (1198831 119883

119899) = 119874 (119899

minus119887) for some 119887 gt 119901 minus 1 (83)

Then for any 120583 gt 0 there exists some positive constant 119896120583

independent of the function 119891 for which

1198641003816100381610038161003816119878119899

(119891) minus 119864119878119899(119891)

1003816100381610038161003816

119901

le 119896120583(119899

1+12058310038171003817100381710038171003817119891

101584010038171003817100381710038171003817

2

+1198991199012 [

[

119899

sum

119895=1

10038161003816100381610038161003816Cov (119891 (119883

1)119891 (119883

119895))10038161003816100381610038161003816]

]

)

(84)

As we can see the lemma assumes association but itworks for negatively associated rvrsquos as well since in the proofit reaches back the result of Newman (see Proposition 15 in[18]) where both types of association are allowed

Let us recall that 120572119899(119909) = radic119899(119865

119899(119909) minus 119864119865

119899(119909)) where

119865119899(119909) = (1119899)sum

119899

119895=1119870((119909 minus 119883

119895)ℎ

119899) It is easy to see that

1198641003816100381610038161003816120572119899

(119909) minus 120572119899(119910)

1003816100381610038161003816

119901

= 119899minus1199012

11986410038161003816100381610038161003816119878119899(119891 (119883

119895)) minus 119864119878

119899(119891 (119883

119895))10038161003816100381610038161003816

119901

(85)

where

119891 (119883119895) = 119870(

119909 minus 119883119895

ℎ119899

) minus 119870(

119910 minus 119883119895

ℎ119899

)

119878119899(119891) =

119899

sum

119895=1

119891 (119883119895)

(86)

With an intent to use Lemma 5 we need 119891 to be boundedfrom above by 1 (which in our case is obvious) and to havebounded 119891

1015840 thus we assume

sup119905isinR

10038161003816100381610038161003816119870

1015840(119905)

10038161003816100381610038161003816

1

ℎ119899

le119862

119870

ℎ119899

lt infin (87)

Now applying inequality (84) we have

1198641003816100381610038161003816120572119899

(119909) minus 120572119899(119910)

1003816100381610038161003816

119901

le119896120583119899minus1199012

1198991+120583

1198622

119870

ℎ2

119899

+1198991199012[

[

119899

sum

119895=1

10038161003816100381610038161003816Cov (119891 (119883

1)119891 (119883

119895))10038161003816100381610038161003816]

]

1199012

(88)

Newman showed in [18] that

Cov (1198911(119883) 119891

2(119884))

= ∬R2119891

1015840

1(119909) 119891

1015840

2(119910)

times [119875 (119883 le 119909 119884 le 119910) minus 119875 (119883 le 119909) 119875 (119884 le 119910)] 119889119909 119889119910

(89)

if 1198911and 119891

2are real valued functions on R having square

integrable derivatives 1198911015840

1and 119891

1015840

2 respectively and provided

that 1198911(119883) 119891

2(119884) have finite secondmoments In light of that

equation and linearity of covariance we get10038161003816100381610038161003816Cov (119891 (119883

1) 119891 (119883

119895))10038161003816100381610038161003816

=

1003816100381610038161003816100381610038161003816∬

R2[119875 (119883

1lt 119909 minus ℎ

119899119904 119883

119895lt 119909 minus ℎ

119899119905)

minus119875 (1198831lt 119909 minus ℎ

119899119904 119883

119895lt 119910 minus ℎ

119899119905)]

minus [119875 (1198831lt 119909 minus ℎ

119899119904) 119875 (119883

119895lt 119909 minus ℎ

119899119905)

minus119875 (1198831lt 119909 minus ℎ

119899119904) 119875 (119883

119895lt 119910 minus ℎ

119899119905)]

minus [119875 (1198831lt 119910 minus ℎ

119899119904 119883

119895lt 119909 minus ℎ

119899119905)

minus119875 (1198831lt 119910 minus ℎ

119899119904 119883

119895lt 119910 minus ℎ

119899119905)]

minus [119875 (1198831lt 119910 minus ℎ

119899119904) 119875 (119883

119895lt 119910 minus ℎ

119899119905)

minus 119875 (1198831lt 119910 minus ℎ

119899119904)

times 119875 (119883119895lt 119909 minus ℎ

119899119905)] 119896 (119904) 119896 (119905) 119889119904 119889119905

10038161003816100381610038161003816100381610038161003816

(90)

Without the loss of generality we may and do assume that119909 gt 119910 thus10038161003816100381610038161003816Cov (119891 (119883

1) 119891 (119883

119895))10038161003816100381610038161003816

=

1003816100381610038161003816100381610038161003816∬

R2119875 (119910 minus ℎ

119899119904 le 119883

1lt 119909 minus ℎ

119899119904

119910 minus ℎ119899119905 le 119883

119895lt 119909 minus ℎ

119899119905)

minus 119875 (119910 minus ℎ119899119904 le 119883

1lt 119909 minus ℎ

119899119904)

times 119875 (119910 minus ℎ119899119905 le 119883

119895lt 119909 minus ℎ

119899119905) 119896 (119904) 119896 (119905) 119889119904 119889119905

1003816100381610038161003816100381610038161003816

(91)

and by triangle inequality we get10038161003816100381610038161003816Cov (119891 (119883

1) 119891 (119883

119895))10038161003816100381610038161003816

le ∬R2[int

119909minusℎ119899119904

119910minusℎ119899119904

int

119909minusℎ119899119905

119910minusℎ119899119905

1198911198831119883119895(119906 V) 119889119906 119889V

+int

119909minusℎ119899119904

119910minusℎ119899119904

1198911198831(119906) 119889119906int

119909minusℎ119899119905

119910minusℎ119899119905

119891119883119895(V) 119889V] 119896 (119904) 119896 (119905) 119889119904 119889119905

(92)

10 ISRN Probability and Statistics

1198911198831119883119895

(119906 V) 1198911198831

(119906) and 119891119883119895

(V) are the joint pdf of [1198831 119883

119895]

and marginal pdfrsquos of rvrsquos1198831and119883

119895 respectively We need

to further assume that 1198621stands for a common upper bound

of 1198911198831

and 1198911198831119883119895

for all 119895 isin N that is

100381710038171003817100381710038171198911198831

10038171003817100381710038171003817le 119862

1

1003817100381710038171003817100381710038171198911198831119883119895

100381710038171003817100381710038171003817le 119862

1forall119895 ge 1 (93)

Then we finally obtain

10038161003816100381610038161003816Cov (119891 (119883

1) 119891 (119883

119895))10038161003816100381610038161003816le 2119862

1

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

2

where = max 1198621 119862

2

1

(94)

On the other hand proceeding like Cai and Roussas in [1] wecan shortly get

10038161003816100381610038161003816Cov (119891 (119883

1) 119891 (119883

119895))10038161003816100381610038161003816

le

100381610038161003816100381610038161003816100381610038161003816

Cov(119870(119909 minus 119883

1

ℎ119899

) 119870(

119909 minus 119883119895

ℎ119899

))

100381610038161003816100381610038161003816100381610038161003816

+

100381610038161003816100381610038161003816100381610038161003816

Cov(119870(119909 minus 119883

1

ℎ119899

) 119870(

119910 minus 119883119895

ℎ119899

))

100381610038161003816100381610038161003816100381610038161003816

+

100381610038161003816100381610038161003816100381610038161003816

Cov(119870(119910 minus 119883

1

ℎ119899

) 119870(

119909 minus 119883119895

ℎ119899

))

100381610038161003816100381610038161003816100381610038161003816

+

100381610038161003816100381610038161003816100381610038161003816

Cov(119870(119910 minus 119883

1

ℎ119899

) 119870(

119910 minus 119883119895

ℎ119899

))

100381610038161003816100381610038161003816100381610038161003816

le 41198622

10038161003816100381610038161003816Cov (119883

1 119883

119895)10038161003816100381610038161003816

13

(95)

where 1198622is a constant relevant to the covariance inequality

for negatively associated rvrsquos (see [13])We now arrive at the following inequality

10038161003816100381610038161003816Cov (119891 (119883

1) 119891 (119883

119895))10038161003816100381610038161003816

le max 1198621 119862

2

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

2

10038161003816100381610038161003816Cov(119883

1 119883

119895)10038161003816100381610038161003816

13

(96)

Assuming |Cov(119891(1198831) 119891(119883

119895))| = 119874(119895

minus119903) and proceeding

similarly to (68) we can get

[

[

119899

sum

119895=1

10038161003816100381610038161003816Cov (119891 (119883

1) 119891 (119883

119895))10038161003816100381610038161003816]

]

1199012

le max 1198621 119862

2 [

[

sum

119895lt|119909minus119910|minus2(1199033)

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

2

+ sum

119895ge|119909minus119910|minus2(1199033)

1

1198951199033]

]

1199012

le 2max 1198621 119862

21003816100381610038161003816119909 minus 119910

1003816100381610038161003816

((119903minus3)119903)119901

(97)

To sum up we obtain the following inequality

1198641003816100381610038161003816120572119899

(119909) minus 120572119899(119910)

1003816100381610038161003816

119901

le 119896120583119899

1+120583minus1199012119862

2

119870

ℎ2

119899

+ 2max 1198621 119862

21003816100381610038161003816119909 minus 119910

1003816100381610038161003816

((119903minus3)119903)119901

le 1198621198991+120583minus1199012 1

ℎ2

119899

+1003816100381610038161003816119909 minus 119910

1003816100381610038161003816

((119903minus3)119903)119901

(98)

where 119862 = 119896120583max1198622

119896 2max119862

1 119862

2 The formula of the

tightness criterion (8) implies that ((119903 minus 3)119903)119901 gt 1 so

119903 gt3119901

119901 minus 1 where 119901 gt 2 (99)

Let us recall the assumptions imposed on the bandwidhtsℎ

119899119899ge1

by Cai and Roussas in [1]B1 lim

119899rarrinfinℎ119899= 0 and ℎ

119899gt 0 for all 119899 isin N

+

B2 lim119899rarrinfin

119899ℎ119899= infin thus ℎ

119899= 119874(1119899

1minus120573) 120573 gt 0

B3 lim119899rarrinfin

119899ℎ4

119899= 0 hence ℎ

119899= 119874(1119899

14+120575) 120575 gt 0

In light of the above

1198991+120583minus1199012 1

ℎ2

119899

= 11989932+120583+2120575minus1199012

(100)

where 120583 gt 0 119901 gt 2 and 0 lt 120575 lt 34 Confrontation with thetightness criterion (8) forces

3

2+ 120583 + 2120575 minus

119901

2lt minus

1

2 (101)

which implies 119901 gt 4 Let us conclude with the followingtheorem

Theorem6 Let 120572119899(119909) = radic119899(119865

119899(119909)minus119864119865

119899(119909)) be the empirical

process built on the kernel estimator of the df119865 for a stationarysequence of negatively associated rvrsquos Assume that conditionsA1 A2 (87) B1 B2 B3 are satisfied Then provided that thetightness criterion (8) holds with 119901 gt 4 it suffices to demand

1003816100381610038161003816Cov (1198801 119880

1+119899)1003816100381610038161003816 = 119874(

1

1198993119901(119901minus1)) (102)

in order to obtain120572

119899(sdot) 997888rarr 119861 (sdot) weakly in 119863 [0 1] (103)

where 119861(sdot) is the zero mean Gaussian process with covariancestructure defined by

1205902(119909 119910) = 119865 (119909 and 119910) minus 119865 (119909) 119865 (119910)

+

infin

sum

119895=1

[ Cov (119868 [1198831le 119909] 119868 [119883

119895+1le 119910])

+ Cov (119868 [1198831le 119910] 119868 [119883

119895+1le 119909])]

(104)

and 119909 119910 isin [0 1]

Proof The remaining covergence of finite-dimensional dis-tributions of 120572

119899(119909) is established in [1]

ISRN Probability and Statistics 11

5 Weak Convergence of theRecursive Kernel-Type Empirical Processunder IID Assumption

The aim of this section is to show weak convergence of theempirical process

120572119899(119909) =

1

radic119899

119899

sum

119895=1

[119870(

119909 minus 119883119895

ℎ119895

) minus 119864119870(

119909 minus 119883119895

ℎ119895

)] (105)

built on iid rvrsquos 119883119895119895ge1

Wewill prove that120572

119899(sdot) convergesweakly to the Brownian

bridge 119861(sdot) with the following covariance structure

1205902(119909 119910) = 119865 (119909 and 119910) minus 119865 (119909) 119865 (119910) for 119909 119910 isin [0 1]

(106)

It turns out that the tightness criterion (8) suffices to reachthe goal Convergence of finite-dimensional distributions of120572

119899(sdot) holds and we shall show itProceeding like Cai and Roussas in [1] we need to show

that for any 119886 119887 isin R

119886120572119899(119909) + 119887120572

119899(119910) 997888rarr 119886119861 (119909) + 119887119861 (119910) in distribution

(107)

Let us introduce the notation

119884119894(119909) = 119870(

119909 minus 119883119894

ℎ119894

) minus 119864119870(119909 minus 119883

119894

ℎ119894

) (108)

and look into the covariance structure of 120572119899(sdot)

Cov (120572119899(119909) 120572

119899(119910))

= Cov( 1

radic119899

119899

sum

119894=1

119884119894(119909)

1

radic119899

119899

sum

119894=1

119884119894(119910))

=1

119899

119899

sum

119894=1

Cov (119884119894(119909) 119884

119894(119910))

=1

119899

119899

sum

119894=1

Cov(119870(119909 minus 119883

119894

ℎ119894

) 119870(119910 minus 119883

119894

ℎ119894

))

(109)

where the second equality follows from assumed indepen-dence of rvrsquos 119883

119894119894ge1

Firstly we observe that each summand converges to119865(119909and

119910) minus 119865(119909)119865(119910) since

Cov(119870(119909 minus 119883

119894

ℎ119894

) 119870(119910 minus 119883

119894

ℎ119894

))

= 119864[119870(119909 minus 119883

119894

ℎ119894

)119870(119910 minus 119883

119894

ℎ119894

)]

minus 119864119870(119909 minus 119883

119894

ℎ119894

)119864119870(119910 minus 119883

119894

ℎ119894

)

(110)

where

119864[119870(119909 minus 119883

119894

ℎ119894

)119870(119910 minus 119883

119894

ℎ119894

)]

= int

119909and119910

minusinfin

119870(119909 minus 119905

ℎ119894

)119870(119910 minus 119905

ℎ119894

)119889119865 (119905)

+ int

119909or119910

119909and119910

119870(119909 minus 119905

ℎ119894

)119870(119910 minus 119905

ℎ119894

)119889119865 (119905)

+ int

infin

119909or119910

119870(119909 minus 119905

ℎ119894

)119870(119910 minus 119905

ℎ119894

)119889119865 (119905)

(111)

and 119865 denotes the common df of the rvrsquos 119883119894119894ge1

In [6] itwas shown that

119864119870(119909 minus 119883

119894

ℎ119894

) 997888rarr 119865 (119909) (112)

and recalling that the kernel function 119870 is a df as well wearrive at the conclusion

Secondly applying Toeplitz lemma we get

Cov (120572119899(119909) 120572

119899(119910)) 997888rarr 119865 (119909 and 119910) minus 119865 (119909) 119865 (119910)

= 1205902(119909 119910) 119899 997888rarr infin

(113)

Thus

Var (119886120572119899(119909) + 119887120572

119899(119910))

= 1198862 Var (120572

119899(119909)) + 2119886119887Cov (120572

119899(119909) 120572

119899(119910))

+ 1198872 Var (120572

119899(119910))

997888rarr 1198862120590

2(119909 119909) + 2119886119887120590

2(119909 119910) + 119887

2120590

2(119910 119910) 119899 997888rarr infin

(114)

We are now on the way to prove that 119886120572119899(119909) + 119887120572

119899(119910) con-

verges in distribution to 119886119861(119909) + 119887119861(119910) sim N(0 1198862120590

2(119909 119909) +

21198861198871205902(119909 119910) + 119887

2120590

2(119910 119910)) which in other words means that

1

radic119899

infin

sum

119894=1

119886119884119894(119909) + 119887119884

119894(119910)

radicVar (119886120572119899(119909) + 119887120572

119899(119910))

997888rarr N (0 1) (115)

In order to obtain (115) we will use Lyapunov condition

lim119899rarrinfin

1

1198632+120575

119899

119899

sum

119894=1

1198641003816100381610038161003816119886119884119894

(119909) + 119887119884119894(119910)

1003816100381610038161003816

2+120575

= 0 (116)

for 120575 = 1 where 1198632

119899= Var(sum119899

119894=1[119886119884

119894(119909) + 119887119884

119894(119910)]) Since

|119870(sdot) minus 119864119870(sdot)| le 2

1198641003816100381610038161003816119886119884119894

(119909) + 119887119884i (119910)1003816100381610038161003816

3

le 8 (1198863+ 119887

3) (117)

Moreover 1198632

119899= 119899Var(119886120572

119899(119909) + 119887120572

119899(119910)) together with (114)

yields

1198633

119899= 119874 (119899

32) (118)

12 ISRN Probability and Statistics

Combining (117) with (118) we obtain

1

1198633

119899

119899

sum

119894=1

1198641003816100381610038161003816119886119884119894

(119909) + 119887119884119894(119910)

1003816100381610038161003816

3

= 119874 (119899minus12

) (119)

which shows that Lyapunov condition holds and completesthe proof of convergence of finite-dimensional distributionsof 120572

119899(119909)We summarize the result of that section in the following

theorem

Theorem 7 Let 120572119899(119909) be the recursive kernel-type empirical

process defined by (105) built on iid rvrsquos 119883119895119895ge1

withmarginal df 119865 If 120572

119899(sdot) satisfies the tightness criterion (8) then

120572119899(sdot) 997888rarr 119861 (sdot) weakly in 119863 [0 1] (120)

where 119861 is the Brownian bridge

Acknowledgment

The author would like to thank the referees for the carefulreading of the paper and helpful remarks

References

[1] Z Cai and G G Roussas ldquoWeak convergence for smoothestimator of a distribution function under negative associationrdquoStochastic Analysis and Applications vol 17 no 2 pp 145ndash1681999

[2] Z Cai andG G Roussas ldquoEfficient Estimation of a DistributionFunction under Qudrant Dependencerdquo Scandinavian Journal ofStatistics vol 25 no 1 pp 211ndash224 1998

[3] ZW Cai and G G Roussas ldquoUniform strong estimation under120572-mixing with ratesrdquo Statistics amp Probability Letters vol 15 no1 pp 47ndash55 1992

[4] G G Roussas ldquoKernel estimates under association stronguniform consistencyrdquo Statistics amp Probability Letters vol 12 no5 pp 393ndash403 1991

[5] Y F Li and S C Yang ldquoUniformly asymptotic normalityof the smooth estimation of the distribution function underassociated samplesrdquo Acta Mathematicae Applicatae Sinica vol28 no 4 pp 639ndash651 2005

[6] Y Li C Wei and S Yang ldquoThe recursive kernel distributionfunction estimator based onnegatively and positively associatedsequencesrdquo Communications in Statistics vol 39 no 20 pp3585ndash3595 2010

[7] H Yu ldquoA Glivenko-Cantelli lemma and weak convergence forempirical processes of associated sequencesrdquo ProbabilityTheoryand Related Fields vol 95 no 3 pp 357ndash370 1993

[8] Q M Shao and H Yu ldquoWeak convergence for weightedempirical processes of dependent sequencesrdquo The Annals ofProbability vol 24 no 4 pp 2098ndash2127 1996

[9] P Billingsley Convergence of Probability Measures Wiley Seriesin Probability and Statistics Probability and Statistics JohnWiley amp Sons New York NY USA 1999

[10] J D Esary F Proschan and D W Walkup ldquoAssociation ofrandom variables with applicationsrdquo Annals of MathematicalStatistics vol 38 pp 1466ndash1474 1967

[11] A Bulinski and A Shashkin Limit Theorems for AssociatedRandom Fields and Related Systems vol 10 of Advanced Serieson Statistical Science and Applied Probability World Scientific2007

[12] S Karlin and Y Rinott ldquoClasses of orderings of measures andrelated correlation inequalities I Multivariate totally positivedistributionsrdquo Journal of Multivariate Analysis vol 10 no 4 pp467ndash498 1980

[13] P Matuła and M Ziemba ldquoGeneralized covariance inequali-tiesrdquo Central European Journal of Mathematics vol 9 no 2 pp281ndash293 2011

[14] P Doukhan and S Louhichi ldquoA new weak dependence condi-tion and applications to moment inequalitiesrdquo Stochastic Proc-esses and their Applications vol 84 no 2 pp 313ndash342 1999

[15] P Matuła ldquoA note on some inequalities for certain classes ofpositively dependent random variablesrdquo Probability and Math-ematical Statistics vol 24 no 1 pp 17ndash26 2004

[16] S Louhichi ldquoWeak convergence for empirical processes of asso-ciated sequencesrdquo Annales de lrsquoInstitut Henri Poincare vol 36no 5 pp 547ndash567 2000

[17] K Joag-Dev and F Proschan ldquoNegative association of randomvariables with applicationsrdquoThe Annals of Statistics vol 11 no1 pp 286ndash295 1983

[18] C M Newman ldquoAsymptotic independence and limit theoremsfor positively and negatively dependent random variablesrdquo inInequalities in Statistics and Probability Y L Tong Ed vol 5pp 127ndash140 Institute ofMathematical StatisticsHayward CalifUSA 1984

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 ISRN Probability and Statistics

The terms obtained in (54) may further be bounded fromabove in the following way

sum

0le119894119896le119895

1198621198942sdot 119862

1198962=

119899minus1

sum

119895=0

119895

sum

119894119896=0

1198621198942119862

1198962

le 119899

119899minus1

sum

119894119896=0

1198621198942119862

1198962= 119899(

119899minus1

sum

119894=0

1198621198942)

2

sum

0le119896119895le119894

1198621198944

=

119899minus1

sum

119894=0

119894

sum

119895119896=0

1198621198944

le

119899minus1

sum

119894=0

(119894 + 1) (119894 + 1) 1198621198944

=

119899minus1

sum

119894=0

(119894 + 1)2119862

1198944

(55)

We thus get the inequality

1198641198784

119899le 4([119899

119899minus1

sum

119894=0

1198621198942]

2

+ 3119899

119899minus1

sum

119894=0

(119894 + 1)2119862

1198944) (56)

We shall now focus on estimating 1198621199032

and 1198621199034 Since 119880

1199051

and 1198801199052

are associated uniformly [0 1] distributed rvrsquos 1198831199051

and 1198831199052

mdashas monotone functions of these rvrsquosmdashare associat-ed as well In order to bound

1198621199032

= sup0le1199051lt1199052le119899 1199052minus1199051=119903

Cov (1198831199051

1198831199052

) (57)

let us notice that from Schwarz inequality

Cov (1198831199051

1198831199052

) = 119864 (119868 [1198801199051

le 119909] 119868 [1198801199052

le 119909]) minus 1199092

le radic1199092 minus 1199092le 119909

(58)

On the other hand invoking inequalities from [13 15]

Cov (1198831199051

1198831199052

)

le sup119909isin[01]

[119875 (1198801199051

le 1199091198801199052

le 119909) minus 119875 (1198801199051

le 119909)119875 (1198801199052

le 119909)]

le 119862 sdot Cov13

(1198801199051

1198801199052

) for associated rvrsquos4 sdot Cov (119880

1199051

1198801199052

) for MTP2rvrsquos

(59)

As a consequence

1198621199032

le min 119909 119862 sdot Cov13

(1198801199051

1198801199052

) for associated rvrsquosmin 119909 4 sdot Cov (119880

1199051

1198801199052

) for MTP2rvrsquos

(60)

where 1199052minus 119905

1= 119903 Still we need the upper bound for 119862

1199034 It

turns out that119862

1199034

= sup1003816100381610038161003816100381610038161003816100381610038161003816

Cov2

prod

119894=1

(119868 [119880119905119894

le 119909] minus 119909)

4

prod

119894=3

(119868 [119880119905119894

le 119909] minus 119909)

1003816100381610038161003816100381610038161003816100381610038161003816

(61)

In other words among all divisions of the group 119883119905119894

119894 isin

1 2 3 4 into two subgroups 1198621199034

attains the biggest valuein case we take two subgroups consisted of two rvrsquos Thesupremum runs over the set 119905

1 119905

2 119905

3 119905

4isin N 0 le 119905

1lt 119905

2lt

1199053lt 119905

4le 119899 and 119905

3minus 119905

2= 119903 Elementary calculation leads to the

following formula

1198621199034

= sup 10038161003816100381610038161003816119875 (119880119905119894

le 119909 119894 isin 1 2 3 4)

minus 119875 (119880119905119894

le 119909 119894 isin 1 2) 119875 (119880119905119894

le 119909 119894 isin 3 4)

minus 119909 [119875 (119880119905119894

le 119909 119894 isin 1 2 3)

minus 119875 (119880119905119894

le 119909 119894 isin 1 2) 119875 (1198801199053

le 119909)

+ 119875 (119880119905119894

le 119909 119894 isin 1 3 4)

minus 119875 (119880119905119894

le 119909 119894 isin 3 4) 119875 (1198801199051

le 119909)

+ 119875 (119880119905119894

le 119909 119894 isin 1 2 4)

minus 119875 (119880119905119894

le 119909 119894 isin 1 2) 119875 (1198801199054

le 119909)

+ 119875 (119880119905119894

le 119909 119894 isin 2 3 4)

minus119875 (119880119905119894

le 119909 119894 isin 3 4) 119875 (1198801199052

le 119909)]

+ 1199092[119875 (119880

1199051

le 1199091198801199053

le 119909)

minus 119875 (1198801199051

le 119909)119875 (1198801199053

le 119909)

+ 119875 (1198801199051

le 1199091198801199054

le 119909)

minus 119875 (1198801199051

le 119909)119875 (1198801199054

le 119909)

+ 119875 (1198801199052

le 1199091198801199053

le 119909)

minus 119875 (1198801199052

le 119909)119875 (1198801199053

le 119909)

+ 119875 (1198801199052

le 1199091198801199054

le 119909)

minus119875 (1198801199052

le 119909)119875 (1198801199052

le 119909)]10038161003816100381610038161003816

le10038161003816100381610038161198771

1003816100381610038161003816 +10038161003816100381610038161198772

1003816100381610038161003816 +10038161003816100381610038161198773

1003816100381610038161003816

(62)

where for the sake of simplicity 1198771 119877

2 and 119877

3are respec-

tively the free coefficient the expression with 119909 and theexpression with 119909

2 Using the Lebowitz inequality (see [15]for instance) and inequalities obtained in [13 15] we arrive at10038161003816100381610038161198771

1003816100381610038161003816 =10038161003816100381610038161003816119875 (119880

119905119894

le 119909 119894 isin 1 2 3 4)

minus119875 (119880119905119894

le 119909 119894 isin 1 2) 119875 (119880119905119894

le 119909 119894 isin 3 4)10038161003816100381610038161003816

le 11986711988011990511198801199053

+ 11986711988011990511198801199054

+ 11986711988011990521198801199053

+ 11986711988011990521198801199054

ISRN Probability and Statistics 7

le

119862(

2

sum

119894=1

4

sum

119895=3

Cov13(119880

119905119894

119880119905119895

)) for associated rvrsquos

4(

2

sum

119894=1

4

sum

119895=3

Cov (119880119905119894

119880119905119895

)) for MTP2rvrsquos

(63)

Analogously we get

10038161003816100381610038161198772

1003816100381610038161003816

le

2119909 sdot 119862(

2

sum

119894=1

4

sum

119895=3

Cov13(119880

119905119894

119880119905119895

)) for associated rvrsquos

2119909 sdot 4(

2

sum

119894=1

4

sum

119895=3

Cov (119880119905119894

119880119905119895

)) for MTP2rvrsquos

10038161003816100381610038161198773

1003816100381610038161003816

le

1199092sdot 119862(

2

sum

119894=1

4

sum

119895=3

Cov13(119880

119905119894

119880119905119895

) ) for associated rvrsquos

1199092sdot 4(

2

sum

119894=1

4

sum

119895=3

Cov (119880119905119894

119880119905119895

)) for MTP2rvrsquos

(64)

Eventually we have

1198621199034

le

119891 (119909) sdot119862(

2

sum

119894=1

4

sum

119895=3

Cov13(119880

119905119894

119880119905119895

)) for associated rvrsquos

119891 (119909) sdot4(

2

sum

119894=1

4

sum

119895=3

Cov (119880119905119894

119880119905119895

)) for MTP2rvrsquos

(65)

where 1199053minus 119905

2= 119903 and 119891(119909) = (1 + 2119909 + 119909

2) for 119909 isin [0 1]

Let us now introduce the following notation

120579119903=

sup119905isinN

Cov (119880119905 119880

119905+119903) for 119903 isin 1 2

1 for 119903 = 0

(66)

and assume it decays powerly at rate 119886 in the following way

120579119903= 119874(

1

(119903 + 1)119886) for 119886 gt 0 119903 isin 0 1 (67)

Let us get back to inequality (56) we can now carry on Atfirst for associated rvrsquos

1198641198784

119899le 4[119899

119899minus1

sum

119903=0

min 119909 119862 sdot Cov13(119880

1199051

1198801199052

)]

2

+ 4 sdot 3119899

119899minus1

sum

119903=0

(119903 + 1)2119891 (119909) sdot 119862(

2

sum

119894=1

4

sum

119895=3

Cov13(119880

119905119894

119880119905119895

))

le 119863 sdot ([119899

119899minus1

sum

119903=0

min 119909 (119903 + 1)minus1198863

]

2

+119899

119899minus1

sum

119903=0

(119903 + 1)2(119903 + 1)

minus1198863)

= 119863 sdot ([119899

119899

sum

119903=1

min 119909 119903minus1198863]

2

+ 119899

119899

sum

119903=1

1199032119903minus1198863

)

le 119863 sdot ([

[

119899 sum

119903lt119909minus3119886

119909 + 119899 sum

119903ge119909minus3119886

1

1199031198863]

]

2

+ 119899

119899

sum

119903=1

1199032minus1198863

)

le 1198631sdot (119899

2119909

2(119886minus3)119886+ 120585)

(68)

where119863 and1198631are constants and

120585 = 119899

119899

sum

119903=1

1199032minus1198863

=

119874(119899) 119886 gt 9

119874 (119899 ln 119899) 119886 = 9

119874 (1198994minus1198863

) 3 lt 119886 lt 9

(69)

It is worth mentioning that in the last inequality of (68) weused the estimate

int

infin

119909

1

119905119901119889119905 sim

1

119909119901minus1for 119901 gt 1 (70)

At the same time in the case of MTP2rvrsquos we get

1198641198784

119899le 119863

2sdot (119899

2119909

2(119886minus1)119886+ 120577) (71)

where1198632is constant and

120577 = 119899

119899

sum

119903=1

1199032minus119886

=

119874(119899) 119886 gt 3

119874 (119899 ln 119899) 119886 = 3

119874 (1198994minus119886

) 1 lt 119886 lt 3

(72)

Let now 120572119899(119909) = (1radic119899)sum

119899

119894=1(119868[119880

119905119894

le 119909] minus 119909) Then

120572119899(119909) minus 120572

119899(119910) =

1

radic119899

119899

sum

119894=1

(119868 [119909 lt 119880119905119894

le 119910] minus (119909 minus 119910))

for 119909 119910 isin [0 1]

(73)

8 ISRN Probability and Statistics

has the fourth moment estimatedmdashin the case of associatedrvrsquosmdashby

119864[120572119899(119909) minus 120572

119899(119910)]

4

= 119864[1

radic119899

119899

sum

119894=1

(119868 [119909 lt 119880119905119894

le 119910] minus (119909 minus 119910))]

4

=1

1198992119864[

119899

sum

119894=1

(119868 [119909 lt 119880119905119894

le 119910] minus (119909 minus 119910))]

4

le 1198631sdot (

1

119899211989921003816100381610038161003816119909 minus 119910

1003816100381610038161003816

2(119886minus3)119886

+120585

1198992)

= 1198631sdot (

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

2(119886minus3)119886

+120585

1198992)

=

119874(119899minus1) 119886 gt 9

119874(ln 119899119899

) 119886 = 9

119874 (1198992minus1198863

) 3 lt 119886 lt 9

(74)

and in the case of MTP2rvrsquos by

119864[120572119899(119909) minus 120572

119899(119910)]

4

le 1198632sdot (

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

2(119886minus1)119886

+120577

1198992)

=

119874(119899minus1) 119886 gt 3

119874(ln 119899119899

) 119886 = 3

119874 (1198992minus119886

) 1 lt 119886 lt 3

(75)

In light of the Shao and Yursquos criterion our process is tight forassociated rvrsquos when 119886 gt 6 and for MTP

2rvrsquos when 119886 gt 2

Let us sum up this result in the following theorem

Theorem 3 Let 120572119899(119909) = (1radic119899)sum

119899

119894=1(119868[119880

119894le 119909] minus 119909) be the

empirical process built on an associated sequence of uniformly[0 1] distributed rvrsquos 119880

119894119894ge1

Let also

120579119903= sup

119905isinNCov (119880

119905 119880

119905+119903) = 119874 ((119903 + 1)

minus119886)

where 119903 isin 0 1 119886 gt 0

(76)

Then 120572119899(119909)

119899ge1is tight for 119886 gt 6 If the rvrsquos 119880

119894119894ge1

are MTP2

then the process is tight for 119886 gt 2

Yu assumed stationarity of 119880119894119894ge1

andsum

infin

119899=111989965+]Cov(119880

0 119880

119899) lt infin for a positive constant ]

thus the rate of decay 119886 gt 7 5 Our result weakens consid-erably these assumptions especially in the case of MTP

2rvrsquos

Louhichi in [16] proposed a different tightness criterioninvolving the so-called bracketing numbers She managedto enhance Yursquos resultmdasheven more than Shao and Yu in[8]mdashsince she proved that it suffices to take 119886 gt 4 to gettightness of the empirical process based on the associated

rvrsquos Nevertheless she kept the assumption of stationarityvalid

In the final analysis our resultrsquos advantage is the absenceof the stationarity assumption and the rate of decay for120579119903remains (up to the authorrsquos knowledge) unimproved for

MTP2rvrsquos

Unfortunately with a view to obtainingweak convergenceof the process in question that is also convergence offinite-dimensional distributions we do not know how tomanage without the assumption of stationarityTherefore weconclude with the following corollary

Corollary 4 Let 120572119899(119909) = (1radic119899)sum

119899

119894=1(119868[119880

119894le 119909] minus 119909) be the

empirical process built on a stationary associated sequence ofuniformly [0 1] distributed rvrsquos 119880

119894119894ge1

Let also

120579119903= Cov (119880

1 119880

1+119903) = 119874 ((119903 + 1)

minus119886)

where 119903 isin 0 1 119886 gt 0

(77)

Then if 119886 gt 6

120572119899(sdot) 997888rarr 119861 (sdot) weakly in 119863 [0 1] (78)

where 119861(sdot) is the zero mean Gaussian process on [0 1] withcovariance structure defined by

1205902(119909 119910) = 119909 and 119910 minus 119909119910

+

infin

sum

119895=1

[ Cov (119868 [1198801le 119909] 119868 [119880

119895+1le 119910])

+ Cov (119868 [1198801le 119910] 119868 [119880

119895+1le 119909])]

(79)

If the rvrsquos 119880119894119894ge1

are MTP2 then it suffices to be 119886 gt 2 in

order to claim the above convergence

Proof It remains to establish convergence of finite-dimen-sional distributions repeating the procedure from [7]

4 Tightness of the Kernel-TypeEmpirical Process

In this section we shall weaken assumption imposed on thecovariance structure of rvrsquos 119883

119895119895ge1

by Cai and Roussas in [1]for the kernel estimator of the df

119865119899(119909) =

1

119899

119899

sum

119895=1

119870(

119909 minus 119883119895

ℎ119899

) (80)

They deal with a stationary sequence of negatively associatedrvrsquos (cf [17]) and need the same condition as Yu [7] that is

1003816100381610038161003816Cov (1198831 119883

1+119899)1003816100381610038161003816 = 119874(

1

11989975+]) (81)

to get tightness of the smooth empirical process (see condi-tion (A4) in [1])

ISRN Probability and Statistics 9

It turns out that it suffices to have

1003816100381610038161003816Cov (1198801 119880

1+119899)1003816100381610038161003816 = 119874(

1

1198993119901(119901minus1)) (82)

where 119901 gt 4 is a positive constant taken from the tightnesscriterion (8) It is easy to see that asymptotically we get therate 3

On the way to prove it we will also take use of aRosenthal-type inequality due to Shao andYu (seeTheorem 2in [8]) we shall recall in the following lemma

Lemma 5 Let 119901 gt 2 and 119891 be a real valued function boundedby 1 with bounded first derivative Suppose that 119883

119899119899ge1

is asequence of stationary and associated rvrsquos such that for 119899 isin N

Cov (1198831 119883

119899) = 119874 (119899

minus119887) for some 119887 gt 119901 minus 1 (83)

Then for any 120583 gt 0 there exists some positive constant 119896120583

independent of the function 119891 for which

1198641003816100381610038161003816119878119899

(119891) minus 119864119878119899(119891)

1003816100381610038161003816

119901

le 119896120583(119899

1+12058310038171003817100381710038171003817119891

101584010038171003817100381710038171003817

2

+1198991199012 [

[

119899

sum

119895=1

10038161003816100381610038161003816Cov (119891 (119883

1)119891 (119883

119895))10038161003816100381610038161003816]

]

)

(84)

As we can see the lemma assumes association but itworks for negatively associated rvrsquos as well since in the proofit reaches back the result of Newman (see Proposition 15 in[18]) where both types of association are allowed

Let us recall that 120572119899(119909) = radic119899(119865

119899(119909) minus 119864119865

119899(119909)) where

119865119899(119909) = (1119899)sum

119899

119895=1119870((119909 minus 119883

119895)ℎ

119899) It is easy to see that

1198641003816100381610038161003816120572119899

(119909) minus 120572119899(119910)

1003816100381610038161003816

119901

= 119899minus1199012

11986410038161003816100381610038161003816119878119899(119891 (119883

119895)) minus 119864119878

119899(119891 (119883

119895))10038161003816100381610038161003816

119901

(85)

where

119891 (119883119895) = 119870(

119909 minus 119883119895

ℎ119899

) minus 119870(

119910 minus 119883119895

ℎ119899

)

119878119899(119891) =

119899

sum

119895=1

119891 (119883119895)

(86)

With an intent to use Lemma 5 we need 119891 to be boundedfrom above by 1 (which in our case is obvious) and to havebounded 119891

1015840 thus we assume

sup119905isinR

10038161003816100381610038161003816119870

1015840(119905)

10038161003816100381610038161003816

1

ℎ119899

le119862

119870

ℎ119899

lt infin (87)

Now applying inequality (84) we have

1198641003816100381610038161003816120572119899

(119909) minus 120572119899(119910)

1003816100381610038161003816

119901

le119896120583119899minus1199012

1198991+120583

1198622

119870

ℎ2

119899

+1198991199012[

[

119899

sum

119895=1

10038161003816100381610038161003816Cov (119891 (119883

1)119891 (119883

119895))10038161003816100381610038161003816]

]

1199012

(88)

Newman showed in [18] that

Cov (1198911(119883) 119891

2(119884))

= ∬R2119891

1015840

1(119909) 119891

1015840

2(119910)

times [119875 (119883 le 119909 119884 le 119910) minus 119875 (119883 le 119909) 119875 (119884 le 119910)] 119889119909 119889119910

(89)

if 1198911and 119891

2are real valued functions on R having square

integrable derivatives 1198911015840

1and 119891

1015840

2 respectively and provided

that 1198911(119883) 119891

2(119884) have finite secondmoments In light of that

equation and linearity of covariance we get10038161003816100381610038161003816Cov (119891 (119883

1) 119891 (119883

119895))10038161003816100381610038161003816

=

1003816100381610038161003816100381610038161003816∬

R2[119875 (119883

1lt 119909 minus ℎ

119899119904 119883

119895lt 119909 minus ℎ

119899119905)

minus119875 (1198831lt 119909 minus ℎ

119899119904 119883

119895lt 119910 minus ℎ

119899119905)]

minus [119875 (1198831lt 119909 minus ℎ

119899119904) 119875 (119883

119895lt 119909 minus ℎ

119899119905)

minus119875 (1198831lt 119909 minus ℎ

119899119904) 119875 (119883

119895lt 119910 minus ℎ

119899119905)]

minus [119875 (1198831lt 119910 minus ℎ

119899119904 119883

119895lt 119909 minus ℎ

119899119905)

minus119875 (1198831lt 119910 minus ℎ

119899119904 119883

119895lt 119910 minus ℎ

119899119905)]

minus [119875 (1198831lt 119910 minus ℎ

119899119904) 119875 (119883

119895lt 119910 minus ℎ

119899119905)

minus 119875 (1198831lt 119910 minus ℎ

119899119904)

times 119875 (119883119895lt 119909 minus ℎ

119899119905)] 119896 (119904) 119896 (119905) 119889119904 119889119905

10038161003816100381610038161003816100381610038161003816

(90)

Without the loss of generality we may and do assume that119909 gt 119910 thus10038161003816100381610038161003816Cov (119891 (119883

1) 119891 (119883

119895))10038161003816100381610038161003816

=

1003816100381610038161003816100381610038161003816∬

R2119875 (119910 minus ℎ

119899119904 le 119883

1lt 119909 minus ℎ

119899119904

119910 minus ℎ119899119905 le 119883

119895lt 119909 minus ℎ

119899119905)

minus 119875 (119910 minus ℎ119899119904 le 119883

1lt 119909 minus ℎ

119899119904)

times 119875 (119910 minus ℎ119899119905 le 119883

119895lt 119909 minus ℎ

119899119905) 119896 (119904) 119896 (119905) 119889119904 119889119905

1003816100381610038161003816100381610038161003816

(91)

and by triangle inequality we get10038161003816100381610038161003816Cov (119891 (119883

1) 119891 (119883

119895))10038161003816100381610038161003816

le ∬R2[int

119909minusℎ119899119904

119910minusℎ119899119904

int

119909minusℎ119899119905

119910minusℎ119899119905

1198911198831119883119895(119906 V) 119889119906 119889V

+int

119909minusℎ119899119904

119910minusℎ119899119904

1198911198831(119906) 119889119906int

119909minusℎ119899119905

119910minusℎ119899119905

119891119883119895(V) 119889V] 119896 (119904) 119896 (119905) 119889119904 119889119905

(92)

10 ISRN Probability and Statistics

1198911198831119883119895

(119906 V) 1198911198831

(119906) and 119891119883119895

(V) are the joint pdf of [1198831 119883

119895]

and marginal pdfrsquos of rvrsquos1198831and119883

119895 respectively We need

to further assume that 1198621stands for a common upper bound

of 1198911198831

and 1198911198831119883119895

for all 119895 isin N that is

100381710038171003817100381710038171198911198831

10038171003817100381710038171003817le 119862

1

1003817100381710038171003817100381710038171198911198831119883119895

100381710038171003817100381710038171003817le 119862

1forall119895 ge 1 (93)

Then we finally obtain

10038161003816100381610038161003816Cov (119891 (119883

1) 119891 (119883

119895))10038161003816100381610038161003816le 2119862

1

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

2

where = max 1198621 119862

2

1

(94)

On the other hand proceeding like Cai and Roussas in [1] wecan shortly get

10038161003816100381610038161003816Cov (119891 (119883

1) 119891 (119883

119895))10038161003816100381610038161003816

le

100381610038161003816100381610038161003816100381610038161003816

Cov(119870(119909 minus 119883

1

ℎ119899

) 119870(

119909 minus 119883119895

ℎ119899

))

100381610038161003816100381610038161003816100381610038161003816

+

100381610038161003816100381610038161003816100381610038161003816

Cov(119870(119909 minus 119883

1

ℎ119899

) 119870(

119910 minus 119883119895

ℎ119899

))

100381610038161003816100381610038161003816100381610038161003816

+

100381610038161003816100381610038161003816100381610038161003816

Cov(119870(119910 minus 119883

1

ℎ119899

) 119870(

119909 minus 119883119895

ℎ119899

))

100381610038161003816100381610038161003816100381610038161003816

+

100381610038161003816100381610038161003816100381610038161003816

Cov(119870(119910 minus 119883

1

ℎ119899

) 119870(

119910 minus 119883119895

ℎ119899

))

100381610038161003816100381610038161003816100381610038161003816

le 41198622

10038161003816100381610038161003816Cov (119883

1 119883

119895)10038161003816100381610038161003816

13

(95)

where 1198622is a constant relevant to the covariance inequality

for negatively associated rvrsquos (see [13])We now arrive at the following inequality

10038161003816100381610038161003816Cov (119891 (119883

1) 119891 (119883

119895))10038161003816100381610038161003816

le max 1198621 119862

2

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

2

10038161003816100381610038161003816Cov(119883

1 119883

119895)10038161003816100381610038161003816

13

(96)

Assuming |Cov(119891(1198831) 119891(119883

119895))| = 119874(119895

minus119903) and proceeding

similarly to (68) we can get

[

[

119899

sum

119895=1

10038161003816100381610038161003816Cov (119891 (119883

1) 119891 (119883

119895))10038161003816100381610038161003816]

]

1199012

le max 1198621 119862

2 [

[

sum

119895lt|119909minus119910|minus2(1199033)

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

2

+ sum

119895ge|119909minus119910|minus2(1199033)

1

1198951199033]

]

1199012

le 2max 1198621 119862

21003816100381610038161003816119909 minus 119910

1003816100381610038161003816

((119903minus3)119903)119901

(97)

To sum up we obtain the following inequality

1198641003816100381610038161003816120572119899

(119909) minus 120572119899(119910)

1003816100381610038161003816

119901

le 119896120583119899

1+120583minus1199012119862

2

119870

ℎ2

119899

+ 2max 1198621 119862

21003816100381610038161003816119909 minus 119910

1003816100381610038161003816

((119903minus3)119903)119901

le 1198621198991+120583minus1199012 1

ℎ2

119899

+1003816100381610038161003816119909 minus 119910

1003816100381610038161003816

((119903minus3)119903)119901

(98)

where 119862 = 119896120583max1198622

119896 2max119862

1 119862

2 The formula of the

tightness criterion (8) implies that ((119903 minus 3)119903)119901 gt 1 so

119903 gt3119901

119901 minus 1 where 119901 gt 2 (99)

Let us recall the assumptions imposed on the bandwidhtsℎ

119899119899ge1

by Cai and Roussas in [1]B1 lim

119899rarrinfinℎ119899= 0 and ℎ

119899gt 0 for all 119899 isin N

+

B2 lim119899rarrinfin

119899ℎ119899= infin thus ℎ

119899= 119874(1119899

1minus120573) 120573 gt 0

B3 lim119899rarrinfin

119899ℎ4

119899= 0 hence ℎ

119899= 119874(1119899

14+120575) 120575 gt 0

In light of the above

1198991+120583minus1199012 1

ℎ2

119899

= 11989932+120583+2120575minus1199012

(100)

where 120583 gt 0 119901 gt 2 and 0 lt 120575 lt 34 Confrontation with thetightness criterion (8) forces

3

2+ 120583 + 2120575 minus

119901

2lt minus

1

2 (101)

which implies 119901 gt 4 Let us conclude with the followingtheorem

Theorem6 Let 120572119899(119909) = radic119899(119865

119899(119909)minus119864119865

119899(119909)) be the empirical

process built on the kernel estimator of the df119865 for a stationarysequence of negatively associated rvrsquos Assume that conditionsA1 A2 (87) B1 B2 B3 are satisfied Then provided that thetightness criterion (8) holds with 119901 gt 4 it suffices to demand

1003816100381610038161003816Cov (1198801 119880

1+119899)1003816100381610038161003816 = 119874(

1

1198993119901(119901minus1)) (102)

in order to obtain120572

119899(sdot) 997888rarr 119861 (sdot) weakly in 119863 [0 1] (103)

where 119861(sdot) is the zero mean Gaussian process with covariancestructure defined by

1205902(119909 119910) = 119865 (119909 and 119910) minus 119865 (119909) 119865 (119910)

+

infin

sum

119895=1

[ Cov (119868 [1198831le 119909] 119868 [119883

119895+1le 119910])

+ Cov (119868 [1198831le 119910] 119868 [119883

119895+1le 119909])]

(104)

and 119909 119910 isin [0 1]

Proof The remaining covergence of finite-dimensional dis-tributions of 120572

119899(119909) is established in [1]

ISRN Probability and Statistics 11

5 Weak Convergence of theRecursive Kernel-Type Empirical Processunder IID Assumption

The aim of this section is to show weak convergence of theempirical process

120572119899(119909) =

1

radic119899

119899

sum

119895=1

[119870(

119909 minus 119883119895

ℎ119895

) minus 119864119870(

119909 minus 119883119895

ℎ119895

)] (105)

built on iid rvrsquos 119883119895119895ge1

Wewill prove that120572

119899(sdot) convergesweakly to the Brownian

bridge 119861(sdot) with the following covariance structure

1205902(119909 119910) = 119865 (119909 and 119910) minus 119865 (119909) 119865 (119910) for 119909 119910 isin [0 1]

(106)

It turns out that the tightness criterion (8) suffices to reachthe goal Convergence of finite-dimensional distributions of120572

119899(sdot) holds and we shall show itProceeding like Cai and Roussas in [1] we need to show

that for any 119886 119887 isin R

119886120572119899(119909) + 119887120572

119899(119910) 997888rarr 119886119861 (119909) + 119887119861 (119910) in distribution

(107)

Let us introduce the notation

119884119894(119909) = 119870(

119909 minus 119883119894

ℎ119894

) minus 119864119870(119909 minus 119883

119894

ℎ119894

) (108)

and look into the covariance structure of 120572119899(sdot)

Cov (120572119899(119909) 120572

119899(119910))

= Cov( 1

radic119899

119899

sum

119894=1

119884119894(119909)

1

radic119899

119899

sum

119894=1

119884119894(119910))

=1

119899

119899

sum

119894=1

Cov (119884119894(119909) 119884

119894(119910))

=1

119899

119899

sum

119894=1

Cov(119870(119909 minus 119883

119894

ℎ119894

) 119870(119910 minus 119883

119894

ℎ119894

))

(109)

where the second equality follows from assumed indepen-dence of rvrsquos 119883

119894119894ge1

Firstly we observe that each summand converges to119865(119909and

119910) minus 119865(119909)119865(119910) since

Cov(119870(119909 minus 119883

119894

ℎ119894

) 119870(119910 minus 119883

119894

ℎ119894

))

= 119864[119870(119909 minus 119883

119894

ℎ119894

)119870(119910 minus 119883

119894

ℎ119894

)]

minus 119864119870(119909 minus 119883

119894

ℎ119894

)119864119870(119910 minus 119883

119894

ℎ119894

)

(110)

where

119864[119870(119909 minus 119883

119894

ℎ119894

)119870(119910 minus 119883

119894

ℎ119894

)]

= int

119909and119910

minusinfin

119870(119909 minus 119905

ℎ119894

)119870(119910 minus 119905

ℎ119894

)119889119865 (119905)

+ int

119909or119910

119909and119910

119870(119909 minus 119905

ℎ119894

)119870(119910 minus 119905

ℎ119894

)119889119865 (119905)

+ int

infin

119909or119910

119870(119909 minus 119905

ℎ119894

)119870(119910 minus 119905

ℎ119894

)119889119865 (119905)

(111)

and 119865 denotes the common df of the rvrsquos 119883119894119894ge1

In [6] itwas shown that

119864119870(119909 minus 119883

119894

ℎ119894

) 997888rarr 119865 (119909) (112)

and recalling that the kernel function 119870 is a df as well wearrive at the conclusion

Secondly applying Toeplitz lemma we get

Cov (120572119899(119909) 120572

119899(119910)) 997888rarr 119865 (119909 and 119910) minus 119865 (119909) 119865 (119910)

= 1205902(119909 119910) 119899 997888rarr infin

(113)

Thus

Var (119886120572119899(119909) + 119887120572

119899(119910))

= 1198862 Var (120572

119899(119909)) + 2119886119887Cov (120572

119899(119909) 120572

119899(119910))

+ 1198872 Var (120572

119899(119910))

997888rarr 1198862120590

2(119909 119909) + 2119886119887120590

2(119909 119910) + 119887

2120590

2(119910 119910) 119899 997888rarr infin

(114)

We are now on the way to prove that 119886120572119899(119909) + 119887120572

119899(119910) con-

verges in distribution to 119886119861(119909) + 119887119861(119910) sim N(0 1198862120590

2(119909 119909) +

21198861198871205902(119909 119910) + 119887

2120590

2(119910 119910)) which in other words means that

1

radic119899

infin

sum

119894=1

119886119884119894(119909) + 119887119884

119894(119910)

radicVar (119886120572119899(119909) + 119887120572

119899(119910))

997888rarr N (0 1) (115)

In order to obtain (115) we will use Lyapunov condition

lim119899rarrinfin

1

1198632+120575

119899

119899

sum

119894=1

1198641003816100381610038161003816119886119884119894

(119909) + 119887119884119894(119910)

1003816100381610038161003816

2+120575

= 0 (116)

for 120575 = 1 where 1198632

119899= Var(sum119899

119894=1[119886119884

119894(119909) + 119887119884

119894(119910)]) Since

|119870(sdot) minus 119864119870(sdot)| le 2

1198641003816100381610038161003816119886119884119894

(119909) + 119887119884i (119910)1003816100381610038161003816

3

le 8 (1198863+ 119887

3) (117)

Moreover 1198632

119899= 119899Var(119886120572

119899(119909) + 119887120572

119899(119910)) together with (114)

yields

1198633

119899= 119874 (119899

32) (118)

12 ISRN Probability and Statistics

Combining (117) with (118) we obtain

1

1198633

119899

119899

sum

119894=1

1198641003816100381610038161003816119886119884119894

(119909) + 119887119884119894(119910)

1003816100381610038161003816

3

= 119874 (119899minus12

) (119)

which shows that Lyapunov condition holds and completesthe proof of convergence of finite-dimensional distributionsof 120572

119899(119909)We summarize the result of that section in the following

theorem

Theorem 7 Let 120572119899(119909) be the recursive kernel-type empirical

process defined by (105) built on iid rvrsquos 119883119895119895ge1

withmarginal df 119865 If 120572

119899(sdot) satisfies the tightness criterion (8) then

120572119899(sdot) 997888rarr 119861 (sdot) weakly in 119863 [0 1] (120)

where 119861 is the Brownian bridge

Acknowledgment

The author would like to thank the referees for the carefulreading of the paper and helpful remarks

References

[1] Z Cai and G G Roussas ldquoWeak convergence for smoothestimator of a distribution function under negative associationrdquoStochastic Analysis and Applications vol 17 no 2 pp 145ndash1681999

[2] Z Cai andG G Roussas ldquoEfficient Estimation of a DistributionFunction under Qudrant Dependencerdquo Scandinavian Journal ofStatistics vol 25 no 1 pp 211ndash224 1998

[3] ZW Cai and G G Roussas ldquoUniform strong estimation under120572-mixing with ratesrdquo Statistics amp Probability Letters vol 15 no1 pp 47ndash55 1992

[4] G G Roussas ldquoKernel estimates under association stronguniform consistencyrdquo Statistics amp Probability Letters vol 12 no5 pp 393ndash403 1991

[5] Y F Li and S C Yang ldquoUniformly asymptotic normalityof the smooth estimation of the distribution function underassociated samplesrdquo Acta Mathematicae Applicatae Sinica vol28 no 4 pp 639ndash651 2005

[6] Y Li C Wei and S Yang ldquoThe recursive kernel distributionfunction estimator based onnegatively and positively associatedsequencesrdquo Communications in Statistics vol 39 no 20 pp3585ndash3595 2010

[7] H Yu ldquoA Glivenko-Cantelli lemma and weak convergence forempirical processes of associated sequencesrdquo ProbabilityTheoryand Related Fields vol 95 no 3 pp 357ndash370 1993

[8] Q M Shao and H Yu ldquoWeak convergence for weightedempirical processes of dependent sequencesrdquo The Annals ofProbability vol 24 no 4 pp 2098ndash2127 1996

[9] P Billingsley Convergence of Probability Measures Wiley Seriesin Probability and Statistics Probability and Statistics JohnWiley amp Sons New York NY USA 1999

[10] J D Esary F Proschan and D W Walkup ldquoAssociation ofrandom variables with applicationsrdquo Annals of MathematicalStatistics vol 38 pp 1466ndash1474 1967

[11] A Bulinski and A Shashkin Limit Theorems for AssociatedRandom Fields and Related Systems vol 10 of Advanced Serieson Statistical Science and Applied Probability World Scientific2007

[12] S Karlin and Y Rinott ldquoClasses of orderings of measures andrelated correlation inequalities I Multivariate totally positivedistributionsrdquo Journal of Multivariate Analysis vol 10 no 4 pp467ndash498 1980

[13] P Matuła and M Ziemba ldquoGeneralized covariance inequali-tiesrdquo Central European Journal of Mathematics vol 9 no 2 pp281ndash293 2011

[14] P Doukhan and S Louhichi ldquoA new weak dependence condi-tion and applications to moment inequalitiesrdquo Stochastic Proc-esses and their Applications vol 84 no 2 pp 313ndash342 1999

[15] P Matuła ldquoA note on some inequalities for certain classes ofpositively dependent random variablesrdquo Probability and Math-ematical Statistics vol 24 no 1 pp 17ndash26 2004

[16] S Louhichi ldquoWeak convergence for empirical processes of asso-ciated sequencesrdquo Annales de lrsquoInstitut Henri Poincare vol 36no 5 pp 547ndash567 2000

[17] K Joag-Dev and F Proschan ldquoNegative association of randomvariables with applicationsrdquoThe Annals of Statistics vol 11 no1 pp 286ndash295 1983

[18] C M Newman ldquoAsymptotic independence and limit theoremsfor positively and negatively dependent random variablesrdquo inInequalities in Statistics and Probability Y L Tong Ed vol 5pp 127ndash140 Institute ofMathematical StatisticsHayward CalifUSA 1984

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

ISRN Probability and Statistics 7

le

119862(

2

sum

119894=1

4

sum

119895=3

Cov13(119880

119905119894

119880119905119895

)) for associated rvrsquos

4(

2

sum

119894=1

4

sum

119895=3

Cov (119880119905119894

119880119905119895

)) for MTP2rvrsquos

(63)

Analogously we get

10038161003816100381610038161198772

1003816100381610038161003816

le

2119909 sdot 119862(

2

sum

119894=1

4

sum

119895=3

Cov13(119880

119905119894

119880119905119895

)) for associated rvrsquos

2119909 sdot 4(

2

sum

119894=1

4

sum

119895=3

Cov (119880119905119894

119880119905119895

)) for MTP2rvrsquos

10038161003816100381610038161198773

1003816100381610038161003816

le

1199092sdot 119862(

2

sum

119894=1

4

sum

119895=3

Cov13(119880

119905119894

119880119905119895

) ) for associated rvrsquos

1199092sdot 4(

2

sum

119894=1

4

sum

119895=3

Cov (119880119905119894

119880119905119895

)) for MTP2rvrsquos

(64)

Eventually we have

1198621199034

le

119891 (119909) sdot119862(

2

sum

119894=1

4

sum

119895=3

Cov13(119880

119905119894

119880119905119895

)) for associated rvrsquos

119891 (119909) sdot4(

2

sum

119894=1

4

sum

119895=3

Cov (119880119905119894

119880119905119895

)) for MTP2rvrsquos

(65)

where 1199053minus 119905

2= 119903 and 119891(119909) = (1 + 2119909 + 119909

2) for 119909 isin [0 1]

Let us now introduce the following notation

120579119903=

sup119905isinN

Cov (119880119905 119880

119905+119903) for 119903 isin 1 2

1 for 119903 = 0

(66)

and assume it decays powerly at rate 119886 in the following way

120579119903= 119874(

1

(119903 + 1)119886) for 119886 gt 0 119903 isin 0 1 (67)

Let us get back to inequality (56) we can now carry on Atfirst for associated rvrsquos

1198641198784

119899le 4[119899

119899minus1

sum

119903=0

min 119909 119862 sdot Cov13(119880

1199051

1198801199052

)]

2

+ 4 sdot 3119899

119899minus1

sum

119903=0

(119903 + 1)2119891 (119909) sdot 119862(

2

sum

119894=1

4

sum

119895=3

Cov13(119880

119905119894

119880119905119895

))

le 119863 sdot ([119899

119899minus1

sum

119903=0

min 119909 (119903 + 1)minus1198863

]

2

+119899

119899minus1

sum

119903=0

(119903 + 1)2(119903 + 1)

minus1198863)

= 119863 sdot ([119899

119899

sum

119903=1

min 119909 119903minus1198863]

2

+ 119899

119899

sum

119903=1

1199032119903minus1198863

)

le 119863 sdot ([

[

119899 sum

119903lt119909minus3119886

119909 + 119899 sum

119903ge119909minus3119886

1

1199031198863]

]

2

+ 119899

119899

sum

119903=1

1199032minus1198863

)

le 1198631sdot (119899

2119909

2(119886minus3)119886+ 120585)

(68)

where119863 and1198631are constants and

120585 = 119899

119899

sum

119903=1

1199032minus1198863

=

119874(119899) 119886 gt 9

119874 (119899 ln 119899) 119886 = 9

119874 (1198994minus1198863

) 3 lt 119886 lt 9

(69)

It is worth mentioning that in the last inequality of (68) weused the estimate

int

infin

119909

1

119905119901119889119905 sim

1

119909119901minus1for 119901 gt 1 (70)

At the same time in the case of MTP2rvrsquos we get

1198641198784

119899le 119863

2sdot (119899

2119909

2(119886minus1)119886+ 120577) (71)

where1198632is constant and

120577 = 119899

119899

sum

119903=1

1199032minus119886

=

119874(119899) 119886 gt 3

119874 (119899 ln 119899) 119886 = 3

119874 (1198994minus119886

) 1 lt 119886 lt 3

(72)

Let now 120572119899(119909) = (1radic119899)sum

119899

119894=1(119868[119880

119905119894

le 119909] minus 119909) Then

120572119899(119909) minus 120572

119899(119910) =

1

radic119899

119899

sum

119894=1

(119868 [119909 lt 119880119905119894

le 119910] minus (119909 minus 119910))

for 119909 119910 isin [0 1]

(73)

8 ISRN Probability and Statistics

has the fourth moment estimatedmdashin the case of associatedrvrsquosmdashby

119864[120572119899(119909) minus 120572

119899(119910)]

4

= 119864[1

radic119899

119899

sum

119894=1

(119868 [119909 lt 119880119905119894

le 119910] minus (119909 minus 119910))]

4

=1

1198992119864[

119899

sum

119894=1

(119868 [119909 lt 119880119905119894

le 119910] minus (119909 minus 119910))]

4

le 1198631sdot (

1

119899211989921003816100381610038161003816119909 minus 119910

1003816100381610038161003816

2(119886minus3)119886

+120585

1198992)

= 1198631sdot (

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

2(119886minus3)119886

+120585

1198992)

=

119874(119899minus1) 119886 gt 9

119874(ln 119899119899

) 119886 = 9

119874 (1198992minus1198863

) 3 lt 119886 lt 9

(74)

and in the case of MTP2rvrsquos by

119864[120572119899(119909) minus 120572

119899(119910)]

4

le 1198632sdot (

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

2(119886minus1)119886

+120577

1198992)

=

119874(119899minus1) 119886 gt 3

119874(ln 119899119899

) 119886 = 3

119874 (1198992minus119886

) 1 lt 119886 lt 3

(75)

In light of the Shao and Yursquos criterion our process is tight forassociated rvrsquos when 119886 gt 6 and for MTP

2rvrsquos when 119886 gt 2

Let us sum up this result in the following theorem

Theorem 3 Let 120572119899(119909) = (1radic119899)sum

119899

119894=1(119868[119880

119894le 119909] minus 119909) be the

empirical process built on an associated sequence of uniformly[0 1] distributed rvrsquos 119880

119894119894ge1

Let also

120579119903= sup

119905isinNCov (119880

119905 119880

119905+119903) = 119874 ((119903 + 1)

minus119886)

where 119903 isin 0 1 119886 gt 0

(76)

Then 120572119899(119909)

119899ge1is tight for 119886 gt 6 If the rvrsquos 119880

119894119894ge1

are MTP2

then the process is tight for 119886 gt 2

Yu assumed stationarity of 119880119894119894ge1

andsum

infin

119899=111989965+]Cov(119880

0 119880

119899) lt infin for a positive constant ]

thus the rate of decay 119886 gt 7 5 Our result weakens consid-erably these assumptions especially in the case of MTP

2rvrsquos

Louhichi in [16] proposed a different tightness criterioninvolving the so-called bracketing numbers She managedto enhance Yursquos resultmdasheven more than Shao and Yu in[8]mdashsince she proved that it suffices to take 119886 gt 4 to gettightness of the empirical process based on the associated

rvrsquos Nevertheless she kept the assumption of stationarityvalid

In the final analysis our resultrsquos advantage is the absenceof the stationarity assumption and the rate of decay for120579119903remains (up to the authorrsquos knowledge) unimproved for

MTP2rvrsquos

Unfortunately with a view to obtainingweak convergenceof the process in question that is also convergence offinite-dimensional distributions we do not know how tomanage without the assumption of stationarityTherefore weconclude with the following corollary

Corollary 4 Let 120572119899(119909) = (1radic119899)sum

119899

119894=1(119868[119880

119894le 119909] minus 119909) be the

empirical process built on a stationary associated sequence ofuniformly [0 1] distributed rvrsquos 119880

119894119894ge1

Let also

120579119903= Cov (119880

1 119880

1+119903) = 119874 ((119903 + 1)

minus119886)

where 119903 isin 0 1 119886 gt 0

(77)

Then if 119886 gt 6

120572119899(sdot) 997888rarr 119861 (sdot) weakly in 119863 [0 1] (78)

where 119861(sdot) is the zero mean Gaussian process on [0 1] withcovariance structure defined by

1205902(119909 119910) = 119909 and 119910 minus 119909119910

+

infin

sum

119895=1

[ Cov (119868 [1198801le 119909] 119868 [119880

119895+1le 119910])

+ Cov (119868 [1198801le 119910] 119868 [119880

119895+1le 119909])]

(79)

If the rvrsquos 119880119894119894ge1

are MTP2 then it suffices to be 119886 gt 2 in

order to claim the above convergence

Proof It remains to establish convergence of finite-dimen-sional distributions repeating the procedure from [7]

4 Tightness of the Kernel-TypeEmpirical Process

In this section we shall weaken assumption imposed on thecovariance structure of rvrsquos 119883

119895119895ge1

by Cai and Roussas in [1]for the kernel estimator of the df

119865119899(119909) =

1

119899

119899

sum

119895=1

119870(

119909 minus 119883119895

ℎ119899

) (80)

They deal with a stationary sequence of negatively associatedrvrsquos (cf [17]) and need the same condition as Yu [7] that is

1003816100381610038161003816Cov (1198831 119883

1+119899)1003816100381610038161003816 = 119874(

1

11989975+]) (81)

to get tightness of the smooth empirical process (see condi-tion (A4) in [1])

ISRN Probability and Statistics 9

It turns out that it suffices to have

1003816100381610038161003816Cov (1198801 119880

1+119899)1003816100381610038161003816 = 119874(

1

1198993119901(119901minus1)) (82)

where 119901 gt 4 is a positive constant taken from the tightnesscriterion (8) It is easy to see that asymptotically we get therate 3

On the way to prove it we will also take use of aRosenthal-type inequality due to Shao andYu (seeTheorem 2in [8]) we shall recall in the following lemma

Lemma 5 Let 119901 gt 2 and 119891 be a real valued function boundedby 1 with bounded first derivative Suppose that 119883

119899119899ge1

is asequence of stationary and associated rvrsquos such that for 119899 isin N

Cov (1198831 119883

119899) = 119874 (119899

minus119887) for some 119887 gt 119901 minus 1 (83)

Then for any 120583 gt 0 there exists some positive constant 119896120583

independent of the function 119891 for which

1198641003816100381610038161003816119878119899

(119891) minus 119864119878119899(119891)

1003816100381610038161003816

119901

le 119896120583(119899

1+12058310038171003817100381710038171003817119891

101584010038171003817100381710038171003817

2

+1198991199012 [

[

119899

sum

119895=1

10038161003816100381610038161003816Cov (119891 (119883

1)119891 (119883

119895))10038161003816100381610038161003816]

]

)

(84)

As we can see the lemma assumes association but itworks for negatively associated rvrsquos as well since in the proofit reaches back the result of Newman (see Proposition 15 in[18]) where both types of association are allowed

Let us recall that 120572119899(119909) = radic119899(119865

119899(119909) minus 119864119865

119899(119909)) where

119865119899(119909) = (1119899)sum

119899

119895=1119870((119909 minus 119883

119895)ℎ

119899) It is easy to see that

1198641003816100381610038161003816120572119899

(119909) minus 120572119899(119910)

1003816100381610038161003816

119901

= 119899minus1199012

11986410038161003816100381610038161003816119878119899(119891 (119883

119895)) minus 119864119878

119899(119891 (119883

119895))10038161003816100381610038161003816

119901

(85)

where

119891 (119883119895) = 119870(

119909 minus 119883119895

ℎ119899

) minus 119870(

119910 minus 119883119895

ℎ119899

)

119878119899(119891) =

119899

sum

119895=1

119891 (119883119895)

(86)

With an intent to use Lemma 5 we need 119891 to be boundedfrom above by 1 (which in our case is obvious) and to havebounded 119891

1015840 thus we assume

sup119905isinR

10038161003816100381610038161003816119870

1015840(119905)

10038161003816100381610038161003816

1

ℎ119899

le119862

119870

ℎ119899

lt infin (87)

Now applying inequality (84) we have

1198641003816100381610038161003816120572119899

(119909) minus 120572119899(119910)

1003816100381610038161003816

119901

le119896120583119899minus1199012

1198991+120583

1198622

119870

ℎ2

119899

+1198991199012[

[

119899

sum

119895=1

10038161003816100381610038161003816Cov (119891 (119883

1)119891 (119883

119895))10038161003816100381610038161003816]

]

1199012

(88)

Newman showed in [18] that

Cov (1198911(119883) 119891

2(119884))

= ∬R2119891

1015840

1(119909) 119891

1015840

2(119910)

times [119875 (119883 le 119909 119884 le 119910) minus 119875 (119883 le 119909) 119875 (119884 le 119910)] 119889119909 119889119910

(89)

if 1198911and 119891

2are real valued functions on R having square

integrable derivatives 1198911015840

1and 119891

1015840

2 respectively and provided

that 1198911(119883) 119891

2(119884) have finite secondmoments In light of that

equation and linearity of covariance we get10038161003816100381610038161003816Cov (119891 (119883

1) 119891 (119883

119895))10038161003816100381610038161003816

=

1003816100381610038161003816100381610038161003816∬

R2[119875 (119883

1lt 119909 minus ℎ

119899119904 119883

119895lt 119909 minus ℎ

119899119905)

minus119875 (1198831lt 119909 minus ℎ

119899119904 119883

119895lt 119910 minus ℎ

119899119905)]

minus [119875 (1198831lt 119909 minus ℎ

119899119904) 119875 (119883

119895lt 119909 minus ℎ

119899119905)

minus119875 (1198831lt 119909 minus ℎ

119899119904) 119875 (119883

119895lt 119910 minus ℎ

119899119905)]

minus [119875 (1198831lt 119910 minus ℎ

119899119904 119883

119895lt 119909 minus ℎ

119899119905)

minus119875 (1198831lt 119910 minus ℎ

119899119904 119883

119895lt 119910 minus ℎ

119899119905)]

minus [119875 (1198831lt 119910 minus ℎ

119899119904) 119875 (119883

119895lt 119910 minus ℎ

119899119905)

minus 119875 (1198831lt 119910 minus ℎ

119899119904)

times 119875 (119883119895lt 119909 minus ℎ

119899119905)] 119896 (119904) 119896 (119905) 119889119904 119889119905

10038161003816100381610038161003816100381610038161003816

(90)

Without the loss of generality we may and do assume that119909 gt 119910 thus10038161003816100381610038161003816Cov (119891 (119883

1) 119891 (119883

119895))10038161003816100381610038161003816

=

1003816100381610038161003816100381610038161003816∬

R2119875 (119910 minus ℎ

119899119904 le 119883

1lt 119909 minus ℎ

119899119904

119910 minus ℎ119899119905 le 119883

119895lt 119909 minus ℎ

119899119905)

minus 119875 (119910 minus ℎ119899119904 le 119883

1lt 119909 minus ℎ

119899119904)

times 119875 (119910 minus ℎ119899119905 le 119883

119895lt 119909 minus ℎ

119899119905) 119896 (119904) 119896 (119905) 119889119904 119889119905

1003816100381610038161003816100381610038161003816

(91)

and by triangle inequality we get10038161003816100381610038161003816Cov (119891 (119883

1) 119891 (119883

119895))10038161003816100381610038161003816

le ∬R2[int

119909minusℎ119899119904

119910minusℎ119899119904

int

119909minusℎ119899119905

119910minusℎ119899119905

1198911198831119883119895(119906 V) 119889119906 119889V

+int

119909minusℎ119899119904

119910minusℎ119899119904

1198911198831(119906) 119889119906int

119909minusℎ119899119905

119910minusℎ119899119905

119891119883119895(V) 119889V] 119896 (119904) 119896 (119905) 119889119904 119889119905

(92)

10 ISRN Probability and Statistics

1198911198831119883119895

(119906 V) 1198911198831

(119906) and 119891119883119895

(V) are the joint pdf of [1198831 119883

119895]

and marginal pdfrsquos of rvrsquos1198831and119883

119895 respectively We need

to further assume that 1198621stands for a common upper bound

of 1198911198831

and 1198911198831119883119895

for all 119895 isin N that is

100381710038171003817100381710038171198911198831

10038171003817100381710038171003817le 119862

1

1003817100381710038171003817100381710038171198911198831119883119895

100381710038171003817100381710038171003817le 119862

1forall119895 ge 1 (93)

Then we finally obtain

10038161003816100381610038161003816Cov (119891 (119883

1) 119891 (119883

119895))10038161003816100381610038161003816le 2119862

1

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

2

where = max 1198621 119862

2

1

(94)

On the other hand proceeding like Cai and Roussas in [1] wecan shortly get

10038161003816100381610038161003816Cov (119891 (119883

1) 119891 (119883

119895))10038161003816100381610038161003816

le

100381610038161003816100381610038161003816100381610038161003816

Cov(119870(119909 minus 119883

1

ℎ119899

) 119870(

119909 minus 119883119895

ℎ119899

))

100381610038161003816100381610038161003816100381610038161003816

+

100381610038161003816100381610038161003816100381610038161003816

Cov(119870(119909 minus 119883

1

ℎ119899

) 119870(

119910 minus 119883119895

ℎ119899

))

100381610038161003816100381610038161003816100381610038161003816

+

100381610038161003816100381610038161003816100381610038161003816

Cov(119870(119910 minus 119883

1

ℎ119899

) 119870(

119909 minus 119883119895

ℎ119899

))

100381610038161003816100381610038161003816100381610038161003816

+

100381610038161003816100381610038161003816100381610038161003816

Cov(119870(119910 minus 119883

1

ℎ119899

) 119870(

119910 minus 119883119895

ℎ119899

))

100381610038161003816100381610038161003816100381610038161003816

le 41198622

10038161003816100381610038161003816Cov (119883

1 119883

119895)10038161003816100381610038161003816

13

(95)

where 1198622is a constant relevant to the covariance inequality

for negatively associated rvrsquos (see [13])We now arrive at the following inequality

10038161003816100381610038161003816Cov (119891 (119883

1) 119891 (119883

119895))10038161003816100381610038161003816

le max 1198621 119862

2

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

2

10038161003816100381610038161003816Cov(119883

1 119883

119895)10038161003816100381610038161003816

13

(96)

Assuming |Cov(119891(1198831) 119891(119883

119895))| = 119874(119895

minus119903) and proceeding

similarly to (68) we can get

[

[

119899

sum

119895=1

10038161003816100381610038161003816Cov (119891 (119883

1) 119891 (119883

119895))10038161003816100381610038161003816]

]

1199012

le max 1198621 119862

2 [

[

sum

119895lt|119909minus119910|minus2(1199033)

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

2

+ sum

119895ge|119909minus119910|minus2(1199033)

1

1198951199033]

]

1199012

le 2max 1198621 119862

21003816100381610038161003816119909 minus 119910

1003816100381610038161003816

((119903minus3)119903)119901

(97)

To sum up we obtain the following inequality

1198641003816100381610038161003816120572119899

(119909) minus 120572119899(119910)

1003816100381610038161003816

119901

le 119896120583119899

1+120583minus1199012119862

2

119870

ℎ2

119899

+ 2max 1198621 119862

21003816100381610038161003816119909 minus 119910

1003816100381610038161003816

((119903minus3)119903)119901

le 1198621198991+120583minus1199012 1

ℎ2

119899

+1003816100381610038161003816119909 minus 119910

1003816100381610038161003816

((119903minus3)119903)119901

(98)

where 119862 = 119896120583max1198622

119896 2max119862

1 119862

2 The formula of the

tightness criterion (8) implies that ((119903 minus 3)119903)119901 gt 1 so

119903 gt3119901

119901 minus 1 where 119901 gt 2 (99)

Let us recall the assumptions imposed on the bandwidhtsℎ

119899119899ge1

by Cai and Roussas in [1]B1 lim

119899rarrinfinℎ119899= 0 and ℎ

119899gt 0 for all 119899 isin N

+

B2 lim119899rarrinfin

119899ℎ119899= infin thus ℎ

119899= 119874(1119899

1minus120573) 120573 gt 0

B3 lim119899rarrinfin

119899ℎ4

119899= 0 hence ℎ

119899= 119874(1119899

14+120575) 120575 gt 0

In light of the above

1198991+120583minus1199012 1

ℎ2

119899

= 11989932+120583+2120575minus1199012

(100)

where 120583 gt 0 119901 gt 2 and 0 lt 120575 lt 34 Confrontation with thetightness criterion (8) forces

3

2+ 120583 + 2120575 minus

119901

2lt minus

1

2 (101)

which implies 119901 gt 4 Let us conclude with the followingtheorem

Theorem6 Let 120572119899(119909) = radic119899(119865

119899(119909)minus119864119865

119899(119909)) be the empirical

process built on the kernel estimator of the df119865 for a stationarysequence of negatively associated rvrsquos Assume that conditionsA1 A2 (87) B1 B2 B3 are satisfied Then provided that thetightness criterion (8) holds with 119901 gt 4 it suffices to demand

1003816100381610038161003816Cov (1198801 119880

1+119899)1003816100381610038161003816 = 119874(

1

1198993119901(119901minus1)) (102)

in order to obtain120572

119899(sdot) 997888rarr 119861 (sdot) weakly in 119863 [0 1] (103)

where 119861(sdot) is the zero mean Gaussian process with covariancestructure defined by

1205902(119909 119910) = 119865 (119909 and 119910) minus 119865 (119909) 119865 (119910)

+

infin

sum

119895=1

[ Cov (119868 [1198831le 119909] 119868 [119883

119895+1le 119910])

+ Cov (119868 [1198831le 119910] 119868 [119883

119895+1le 119909])]

(104)

and 119909 119910 isin [0 1]

Proof The remaining covergence of finite-dimensional dis-tributions of 120572

119899(119909) is established in [1]

ISRN Probability and Statistics 11

5 Weak Convergence of theRecursive Kernel-Type Empirical Processunder IID Assumption

The aim of this section is to show weak convergence of theempirical process

120572119899(119909) =

1

radic119899

119899

sum

119895=1

[119870(

119909 minus 119883119895

ℎ119895

) minus 119864119870(

119909 minus 119883119895

ℎ119895

)] (105)

built on iid rvrsquos 119883119895119895ge1

Wewill prove that120572

119899(sdot) convergesweakly to the Brownian

bridge 119861(sdot) with the following covariance structure

1205902(119909 119910) = 119865 (119909 and 119910) minus 119865 (119909) 119865 (119910) for 119909 119910 isin [0 1]

(106)

It turns out that the tightness criterion (8) suffices to reachthe goal Convergence of finite-dimensional distributions of120572

119899(sdot) holds and we shall show itProceeding like Cai and Roussas in [1] we need to show

that for any 119886 119887 isin R

119886120572119899(119909) + 119887120572

119899(119910) 997888rarr 119886119861 (119909) + 119887119861 (119910) in distribution

(107)

Let us introduce the notation

119884119894(119909) = 119870(

119909 minus 119883119894

ℎ119894

) minus 119864119870(119909 minus 119883

119894

ℎ119894

) (108)

and look into the covariance structure of 120572119899(sdot)

Cov (120572119899(119909) 120572

119899(119910))

= Cov( 1

radic119899

119899

sum

119894=1

119884119894(119909)

1

radic119899

119899

sum

119894=1

119884119894(119910))

=1

119899

119899

sum

119894=1

Cov (119884119894(119909) 119884

119894(119910))

=1

119899

119899

sum

119894=1

Cov(119870(119909 minus 119883

119894

ℎ119894

) 119870(119910 minus 119883

119894

ℎ119894

))

(109)

where the second equality follows from assumed indepen-dence of rvrsquos 119883

119894119894ge1

Firstly we observe that each summand converges to119865(119909and

119910) minus 119865(119909)119865(119910) since

Cov(119870(119909 minus 119883

119894

ℎ119894

) 119870(119910 minus 119883

119894

ℎ119894

))

= 119864[119870(119909 minus 119883

119894

ℎ119894

)119870(119910 minus 119883

119894

ℎ119894

)]

minus 119864119870(119909 minus 119883

119894

ℎ119894

)119864119870(119910 minus 119883

119894

ℎ119894

)

(110)

where

119864[119870(119909 minus 119883

119894

ℎ119894

)119870(119910 minus 119883

119894

ℎ119894

)]

= int

119909and119910

minusinfin

119870(119909 minus 119905

ℎ119894

)119870(119910 minus 119905

ℎ119894

)119889119865 (119905)

+ int

119909or119910

119909and119910

119870(119909 minus 119905

ℎ119894

)119870(119910 minus 119905

ℎ119894

)119889119865 (119905)

+ int

infin

119909or119910

119870(119909 minus 119905

ℎ119894

)119870(119910 minus 119905

ℎ119894

)119889119865 (119905)

(111)

and 119865 denotes the common df of the rvrsquos 119883119894119894ge1

In [6] itwas shown that

119864119870(119909 minus 119883

119894

ℎ119894

) 997888rarr 119865 (119909) (112)

and recalling that the kernel function 119870 is a df as well wearrive at the conclusion

Secondly applying Toeplitz lemma we get

Cov (120572119899(119909) 120572

119899(119910)) 997888rarr 119865 (119909 and 119910) minus 119865 (119909) 119865 (119910)

= 1205902(119909 119910) 119899 997888rarr infin

(113)

Thus

Var (119886120572119899(119909) + 119887120572

119899(119910))

= 1198862 Var (120572

119899(119909)) + 2119886119887Cov (120572

119899(119909) 120572

119899(119910))

+ 1198872 Var (120572

119899(119910))

997888rarr 1198862120590

2(119909 119909) + 2119886119887120590

2(119909 119910) + 119887

2120590

2(119910 119910) 119899 997888rarr infin

(114)

We are now on the way to prove that 119886120572119899(119909) + 119887120572

119899(119910) con-

verges in distribution to 119886119861(119909) + 119887119861(119910) sim N(0 1198862120590

2(119909 119909) +

21198861198871205902(119909 119910) + 119887

2120590

2(119910 119910)) which in other words means that

1

radic119899

infin

sum

119894=1

119886119884119894(119909) + 119887119884

119894(119910)

radicVar (119886120572119899(119909) + 119887120572

119899(119910))

997888rarr N (0 1) (115)

In order to obtain (115) we will use Lyapunov condition

lim119899rarrinfin

1

1198632+120575

119899

119899

sum

119894=1

1198641003816100381610038161003816119886119884119894

(119909) + 119887119884119894(119910)

1003816100381610038161003816

2+120575

= 0 (116)

for 120575 = 1 where 1198632

119899= Var(sum119899

119894=1[119886119884

119894(119909) + 119887119884

119894(119910)]) Since

|119870(sdot) minus 119864119870(sdot)| le 2

1198641003816100381610038161003816119886119884119894

(119909) + 119887119884i (119910)1003816100381610038161003816

3

le 8 (1198863+ 119887

3) (117)

Moreover 1198632

119899= 119899Var(119886120572

119899(119909) + 119887120572

119899(119910)) together with (114)

yields

1198633

119899= 119874 (119899

32) (118)

12 ISRN Probability and Statistics

Combining (117) with (118) we obtain

1

1198633

119899

119899

sum

119894=1

1198641003816100381610038161003816119886119884119894

(119909) + 119887119884119894(119910)

1003816100381610038161003816

3

= 119874 (119899minus12

) (119)

which shows that Lyapunov condition holds and completesthe proof of convergence of finite-dimensional distributionsof 120572

119899(119909)We summarize the result of that section in the following

theorem

Theorem 7 Let 120572119899(119909) be the recursive kernel-type empirical

process defined by (105) built on iid rvrsquos 119883119895119895ge1

withmarginal df 119865 If 120572

119899(sdot) satisfies the tightness criterion (8) then

120572119899(sdot) 997888rarr 119861 (sdot) weakly in 119863 [0 1] (120)

where 119861 is the Brownian bridge

Acknowledgment

The author would like to thank the referees for the carefulreading of the paper and helpful remarks

References

[1] Z Cai and G G Roussas ldquoWeak convergence for smoothestimator of a distribution function under negative associationrdquoStochastic Analysis and Applications vol 17 no 2 pp 145ndash1681999

[2] Z Cai andG G Roussas ldquoEfficient Estimation of a DistributionFunction under Qudrant Dependencerdquo Scandinavian Journal ofStatistics vol 25 no 1 pp 211ndash224 1998

[3] ZW Cai and G G Roussas ldquoUniform strong estimation under120572-mixing with ratesrdquo Statistics amp Probability Letters vol 15 no1 pp 47ndash55 1992

[4] G G Roussas ldquoKernel estimates under association stronguniform consistencyrdquo Statistics amp Probability Letters vol 12 no5 pp 393ndash403 1991

[5] Y F Li and S C Yang ldquoUniformly asymptotic normalityof the smooth estimation of the distribution function underassociated samplesrdquo Acta Mathematicae Applicatae Sinica vol28 no 4 pp 639ndash651 2005

[6] Y Li C Wei and S Yang ldquoThe recursive kernel distributionfunction estimator based onnegatively and positively associatedsequencesrdquo Communications in Statistics vol 39 no 20 pp3585ndash3595 2010

[7] H Yu ldquoA Glivenko-Cantelli lemma and weak convergence forempirical processes of associated sequencesrdquo ProbabilityTheoryand Related Fields vol 95 no 3 pp 357ndash370 1993

[8] Q M Shao and H Yu ldquoWeak convergence for weightedempirical processes of dependent sequencesrdquo The Annals ofProbability vol 24 no 4 pp 2098ndash2127 1996

[9] P Billingsley Convergence of Probability Measures Wiley Seriesin Probability and Statistics Probability and Statistics JohnWiley amp Sons New York NY USA 1999

[10] J D Esary F Proschan and D W Walkup ldquoAssociation ofrandom variables with applicationsrdquo Annals of MathematicalStatistics vol 38 pp 1466ndash1474 1967

[11] A Bulinski and A Shashkin Limit Theorems for AssociatedRandom Fields and Related Systems vol 10 of Advanced Serieson Statistical Science and Applied Probability World Scientific2007

[12] S Karlin and Y Rinott ldquoClasses of orderings of measures andrelated correlation inequalities I Multivariate totally positivedistributionsrdquo Journal of Multivariate Analysis vol 10 no 4 pp467ndash498 1980

[13] P Matuła and M Ziemba ldquoGeneralized covariance inequali-tiesrdquo Central European Journal of Mathematics vol 9 no 2 pp281ndash293 2011

[14] P Doukhan and S Louhichi ldquoA new weak dependence condi-tion and applications to moment inequalitiesrdquo Stochastic Proc-esses and their Applications vol 84 no 2 pp 313ndash342 1999

[15] P Matuła ldquoA note on some inequalities for certain classes ofpositively dependent random variablesrdquo Probability and Math-ematical Statistics vol 24 no 1 pp 17ndash26 2004

[16] S Louhichi ldquoWeak convergence for empirical processes of asso-ciated sequencesrdquo Annales de lrsquoInstitut Henri Poincare vol 36no 5 pp 547ndash567 2000

[17] K Joag-Dev and F Proschan ldquoNegative association of randomvariables with applicationsrdquoThe Annals of Statistics vol 11 no1 pp 286ndash295 1983

[18] C M Newman ldquoAsymptotic independence and limit theoremsfor positively and negatively dependent random variablesrdquo inInequalities in Statistics and Probability Y L Tong Ed vol 5pp 127ndash140 Institute ofMathematical StatisticsHayward CalifUSA 1984

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 ISRN Probability and Statistics

has the fourth moment estimatedmdashin the case of associatedrvrsquosmdashby

119864[120572119899(119909) minus 120572

119899(119910)]

4

= 119864[1

radic119899

119899

sum

119894=1

(119868 [119909 lt 119880119905119894

le 119910] minus (119909 minus 119910))]

4

=1

1198992119864[

119899

sum

119894=1

(119868 [119909 lt 119880119905119894

le 119910] minus (119909 minus 119910))]

4

le 1198631sdot (

1

119899211989921003816100381610038161003816119909 minus 119910

1003816100381610038161003816

2(119886minus3)119886

+120585

1198992)

= 1198631sdot (

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

2(119886minus3)119886

+120585

1198992)

=

119874(119899minus1) 119886 gt 9

119874(ln 119899119899

) 119886 = 9

119874 (1198992minus1198863

) 3 lt 119886 lt 9

(74)

and in the case of MTP2rvrsquos by

119864[120572119899(119909) minus 120572

119899(119910)]

4

le 1198632sdot (

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

2(119886minus1)119886

+120577

1198992)

=

119874(119899minus1) 119886 gt 3

119874(ln 119899119899

) 119886 = 3

119874 (1198992minus119886

) 1 lt 119886 lt 3

(75)

In light of the Shao and Yursquos criterion our process is tight forassociated rvrsquos when 119886 gt 6 and for MTP

2rvrsquos when 119886 gt 2

Let us sum up this result in the following theorem

Theorem 3 Let 120572119899(119909) = (1radic119899)sum

119899

119894=1(119868[119880

119894le 119909] minus 119909) be the

empirical process built on an associated sequence of uniformly[0 1] distributed rvrsquos 119880

119894119894ge1

Let also

120579119903= sup

119905isinNCov (119880

119905 119880

119905+119903) = 119874 ((119903 + 1)

minus119886)

where 119903 isin 0 1 119886 gt 0

(76)

Then 120572119899(119909)

119899ge1is tight for 119886 gt 6 If the rvrsquos 119880

119894119894ge1

are MTP2

then the process is tight for 119886 gt 2

Yu assumed stationarity of 119880119894119894ge1

andsum

infin

119899=111989965+]Cov(119880

0 119880

119899) lt infin for a positive constant ]

thus the rate of decay 119886 gt 7 5 Our result weakens consid-erably these assumptions especially in the case of MTP

2rvrsquos

Louhichi in [16] proposed a different tightness criterioninvolving the so-called bracketing numbers She managedto enhance Yursquos resultmdasheven more than Shao and Yu in[8]mdashsince she proved that it suffices to take 119886 gt 4 to gettightness of the empirical process based on the associated

rvrsquos Nevertheless she kept the assumption of stationarityvalid

In the final analysis our resultrsquos advantage is the absenceof the stationarity assumption and the rate of decay for120579119903remains (up to the authorrsquos knowledge) unimproved for

MTP2rvrsquos

Unfortunately with a view to obtainingweak convergenceof the process in question that is also convergence offinite-dimensional distributions we do not know how tomanage without the assumption of stationarityTherefore weconclude with the following corollary

Corollary 4 Let 120572119899(119909) = (1radic119899)sum

119899

119894=1(119868[119880

119894le 119909] minus 119909) be the

empirical process built on a stationary associated sequence ofuniformly [0 1] distributed rvrsquos 119880

119894119894ge1

Let also

120579119903= Cov (119880

1 119880

1+119903) = 119874 ((119903 + 1)

minus119886)

where 119903 isin 0 1 119886 gt 0

(77)

Then if 119886 gt 6

120572119899(sdot) 997888rarr 119861 (sdot) weakly in 119863 [0 1] (78)

where 119861(sdot) is the zero mean Gaussian process on [0 1] withcovariance structure defined by

1205902(119909 119910) = 119909 and 119910 minus 119909119910

+

infin

sum

119895=1

[ Cov (119868 [1198801le 119909] 119868 [119880

119895+1le 119910])

+ Cov (119868 [1198801le 119910] 119868 [119880

119895+1le 119909])]

(79)

If the rvrsquos 119880119894119894ge1

are MTP2 then it suffices to be 119886 gt 2 in

order to claim the above convergence

Proof It remains to establish convergence of finite-dimen-sional distributions repeating the procedure from [7]

4 Tightness of the Kernel-TypeEmpirical Process

In this section we shall weaken assumption imposed on thecovariance structure of rvrsquos 119883

119895119895ge1

by Cai and Roussas in [1]for the kernel estimator of the df

119865119899(119909) =

1

119899

119899

sum

119895=1

119870(

119909 minus 119883119895

ℎ119899

) (80)

They deal with a stationary sequence of negatively associatedrvrsquos (cf [17]) and need the same condition as Yu [7] that is

1003816100381610038161003816Cov (1198831 119883

1+119899)1003816100381610038161003816 = 119874(

1

11989975+]) (81)

to get tightness of the smooth empirical process (see condi-tion (A4) in [1])

ISRN Probability and Statistics 9

It turns out that it suffices to have

1003816100381610038161003816Cov (1198801 119880

1+119899)1003816100381610038161003816 = 119874(

1

1198993119901(119901minus1)) (82)

where 119901 gt 4 is a positive constant taken from the tightnesscriterion (8) It is easy to see that asymptotically we get therate 3

On the way to prove it we will also take use of aRosenthal-type inequality due to Shao andYu (seeTheorem 2in [8]) we shall recall in the following lemma

Lemma 5 Let 119901 gt 2 and 119891 be a real valued function boundedby 1 with bounded first derivative Suppose that 119883

119899119899ge1

is asequence of stationary and associated rvrsquos such that for 119899 isin N

Cov (1198831 119883

119899) = 119874 (119899

minus119887) for some 119887 gt 119901 minus 1 (83)

Then for any 120583 gt 0 there exists some positive constant 119896120583

independent of the function 119891 for which

1198641003816100381610038161003816119878119899

(119891) minus 119864119878119899(119891)

1003816100381610038161003816

119901

le 119896120583(119899

1+12058310038171003817100381710038171003817119891

101584010038171003817100381710038171003817

2

+1198991199012 [

[

119899

sum

119895=1

10038161003816100381610038161003816Cov (119891 (119883

1)119891 (119883

119895))10038161003816100381610038161003816]

]

)

(84)

As we can see the lemma assumes association but itworks for negatively associated rvrsquos as well since in the proofit reaches back the result of Newman (see Proposition 15 in[18]) where both types of association are allowed

Let us recall that 120572119899(119909) = radic119899(119865

119899(119909) minus 119864119865

119899(119909)) where

119865119899(119909) = (1119899)sum

119899

119895=1119870((119909 minus 119883

119895)ℎ

119899) It is easy to see that

1198641003816100381610038161003816120572119899

(119909) minus 120572119899(119910)

1003816100381610038161003816

119901

= 119899minus1199012

11986410038161003816100381610038161003816119878119899(119891 (119883

119895)) minus 119864119878

119899(119891 (119883

119895))10038161003816100381610038161003816

119901

(85)

where

119891 (119883119895) = 119870(

119909 minus 119883119895

ℎ119899

) minus 119870(

119910 minus 119883119895

ℎ119899

)

119878119899(119891) =

119899

sum

119895=1

119891 (119883119895)

(86)

With an intent to use Lemma 5 we need 119891 to be boundedfrom above by 1 (which in our case is obvious) and to havebounded 119891

1015840 thus we assume

sup119905isinR

10038161003816100381610038161003816119870

1015840(119905)

10038161003816100381610038161003816

1

ℎ119899

le119862

119870

ℎ119899

lt infin (87)

Now applying inequality (84) we have

1198641003816100381610038161003816120572119899

(119909) minus 120572119899(119910)

1003816100381610038161003816

119901

le119896120583119899minus1199012

1198991+120583

1198622

119870

ℎ2

119899

+1198991199012[

[

119899

sum

119895=1

10038161003816100381610038161003816Cov (119891 (119883

1)119891 (119883

119895))10038161003816100381610038161003816]

]

1199012

(88)

Newman showed in [18] that

Cov (1198911(119883) 119891

2(119884))

= ∬R2119891

1015840

1(119909) 119891

1015840

2(119910)

times [119875 (119883 le 119909 119884 le 119910) minus 119875 (119883 le 119909) 119875 (119884 le 119910)] 119889119909 119889119910

(89)

if 1198911and 119891

2are real valued functions on R having square

integrable derivatives 1198911015840

1and 119891

1015840

2 respectively and provided

that 1198911(119883) 119891

2(119884) have finite secondmoments In light of that

equation and linearity of covariance we get10038161003816100381610038161003816Cov (119891 (119883

1) 119891 (119883

119895))10038161003816100381610038161003816

=

1003816100381610038161003816100381610038161003816∬

R2[119875 (119883

1lt 119909 minus ℎ

119899119904 119883

119895lt 119909 minus ℎ

119899119905)

minus119875 (1198831lt 119909 minus ℎ

119899119904 119883

119895lt 119910 minus ℎ

119899119905)]

minus [119875 (1198831lt 119909 minus ℎ

119899119904) 119875 (119883

119895lt 119909 minus ℎ

119899119905)

minus119875 (1198831lt 119909 minus ℎ

119899119904) 119875 (119883

119895lt 119910 minus ℎ

119899119905)]

minus [119875 (1198831lt 119910 minus ℎ

119899119904 119883

119895lt 119909 minus ℎ

119899119905)

minus119875 (1198831lt 119910 minus ℎ

119899119904 119883

119895lt 119910 minus ℎ

119899119905)]

minus [119875 (1198831lt 119910 minus ℎ

119899119904) 119875 (119883

119895lt 119910 minus ℎ

119899119905)

minus 119875 (1198831lt 119910 minus ℎ

119899119904)

times 119875 (119883119895lt 119909 minus ℎ

119899119905)] 119896 (119904) 119896 (119905) 119889119904 119889119905

10038161003816100381610038161003816100381610038161003816

(90)

Without the loss of generality we may and do assume that119909 gt 119910 thus10038161003816100381610038161003816Cov (119891 (119883

1) 119891 (119883

119895))10038161003816100381610038161003816

=

1003816100381610038161003816100381610038161003816∬

R2119875 (119910 minus ℎ

119899119904 le 119883

1lt 119909 minus ℎ

119899119904

119910 minus ℎ119899119905 le 119883

119895lt 119909 minus ℎ

119899119905)

minus 119875 (119910 minus ℎ119899119904 le 119883

1lt 119909 minus ℎ

119899119904)

times 119875 (119910 minus ℎ119899119905 le 119883

119895lt 119909 minus ℎ

119899119905) 119896 (119904) 119896 (119905) 119889119904 119889119905

1003816100381610038161003816100381610038161003816

(91)

and by triangle inequality we get10038161003816100381610038161003816Cov (119891 (119883

1) 119891 (119883

119895))10038161003816100381610038161003816

le ∬R2[int

119909minusℎ119899119904

119910minusℎ119899119904

int

119909minusℎ119899119905

119910minusℎ119899119905

1198911198831119883119895(119906 V) 119889119906 119889V

+int

119909minusℎ119899119904

119910minusℎ119899119904

1198911198831(119906) 119889119906int

119909minusℎ119899119905

119910minusℎ119899119905

119891119883119895(V) 119889V] 119896 (119904) 119896 (119905) 119889119904 119889119905

(92)

10 ISRN Probability and Statistics

1198911198831119883119895

(119906 V) 1198911198831

(119906) and 119891119883119895

(V) are the joint pdf of [1198831 119883

119895]

and marginal pdfrsquos of rvrsquos1198831and119883

119895 respectively We need

to further assume that 1198621stands for a common upper bound

of 1198911198831

and 1198911198831119883119895

for all 119895 isin N that is

100381710038171003817100381710038171198911198831

10038171003817100381710038171003817le 119862

1

1003817100381710038171003817100381710038171198911198831119883119895

100381710038171003817100381710038171003817le 119862

1forall119895 ge 1 (93)

Then we finally obtain

10038161003816100381610038161003816Cov (119891 (119883

1) 119891 (119883

119895))10038161003816100381610038161003816le 2119862

1

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

2

where = max 1198621 119862

2

1

(94)

On the other hand proceeding like Cai and Roussas in [1] wecan shortly get

10038161003816100381610038161003816Cov (119891 (119883

1) 119891 (119883

119895))10038161003816100381610038161003816

le

100381610038161003816100381610038161003816100381610038161003816

Cov(119870(119909 minus 119883

1

ℎ119899

) 119870(

119909 minus 119883119895

ℎ119899

))

100381610038161003816100381610038161003816100381610038161003816

+

100381610038161003816100381610038161003816100381610038161003816

Cov(119870(119909 minus 119883

1

ℎ119899

) 119870(

119910 minus 119883119895

ℎ119899

))

100381610038161003816100381610038161003816100381610038161003816

+

100381610038161003816100381610038161003816100381610038161003816

Cov(119870(119910 minus 119883

1

ℎ119899

) 119870(

119909 minus 119883119895

ℎ119899

))

100381610038161003816100381610038161003816100381610038161003816

+

100381610038161003816100381610038161003816100381610038161003816

Cov(119870(119910 minus 119883

1

ℎ119899

) 119870(

119910 minus 119883119895

ℎ119899

))

100381610038161003816100381610038161003816100381610038161003816

le 41198622

10038161003816100381610038161003816Cov (119883

1 119883

119895)10038161003816100381610038161003816

13

(95)

where 1198622is a constant relevant to the covariance inequality

for negatively associated rvrsquos (see [13])We now arrive at the following inequality

10038161003816100381610038161003816Cov (119891 (119883

1) 119891 (119883

119895))10038161003816100381610038161003816

le max 1198621 119862

2

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

2

10038161003816100381610038161003816Cov(119883

1 119883

119895)10038161003816100381610038161003816

13

(96)

Assuming |Cov(119891(1198831) 119891(119883

119895))| = 119874(119895

minus119903) and proceeding

similarly to (68) we can get

[

[

119899

sum

119895=1

10038161003816100381610038161003816Cov (119891 (119883

1) 119891 (119883

119895))10038161003816100381610038161003816]

]

1199012

le max 1198621 119862

2 [

[

sum

119895lt|119909minus119910|minus2(1199033)

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

2

+ sum

119895ge|119909minus119910|minus2(1199033)

1

1198951199033]

]

1199012

le 2max 1198621 119862

21003816100381610038161003816119909 minus 119910

1003816100381610038161003816

((119903minus3)119903)119901

(97)

To sum up we obtain the following inequality

1198641003816100381610038161003816120572119899

(119909) minus 120572119899(119910)

1003816100381610038161003816

119901

le 119896120583119899

1+120583minus1199012119862

2

119870

ℎ2

119899

+ 2max 1198621 119862

21003816100381610038161003816119909 minus 119910

1003816100381610038161003816

((119903minus3)119903)119901

le 1198621198991+120583minus1199012 1

ℎ2

119899

+1003816100381610038161003816119909 minus 119910

1003816100381610038161003816

((119903minus3)119903)119901

(98)

where 119862 = 119896120583max1198622

119896 2max119862

1 119862

2 The formula of the

tightness criterion (8) implies that ((119903 minus 3)119903)119901 gt 1 so

119903 gt3119901

119901 minus 1 where 119901 gt 2 (99)

Let us recall the assumptions imposed on the bandwidhtsℎ

119899119899ge1

by Cai and Roussas in [1]B1 lim

119899rarrinfinℎ119899= 0 and ℎ

119899gt 0 for all 119899 isin N

+

B2 lim119899rarrinfin

119899ℎ119899= infin thus ℎ

119899= 119874(1119899

1minus120573) 120573 gt 0

B3 lim119899rarrinfin

119899ℎ4

119899= 0 hence ℎ

119899= 119874(1119899

14+120575) 120575 gt 0

In light of the above

1198991+120583minus1199012 1

ℎ2

119899

= 11989932+120583+2120575minus1199012

(100)

where 120583 gt 0 119901 gt 2 and 0 lt 120575 lt 34 Confrontation with thetightness criterion (8) forces

3

2+ 120583 + 2120575 minus

119901

2lt minus

1

2 (101)

which implies 119901 gt 4 Let us conclude with the followingtheorem

Theorem6 Let 120572119899(119909) = radic119899(119865

119899(119909)minus119864119865

119899(119909)) be the empirical

process built on the kernel estimator of the df119865 for a stationarysequence of negatively associated rvrsquos Assume that conditionsA1 A2 (87) B1 B2 B3 are satisfied Then provided that thetightness criterion (8) holds with 119901 gt 4 it suffices to demand

1003816100381610038161003816Cov (1198801 119880

1+119899)1003816100381610038161003816 = 119874(

1

1198993119901(119901minus1)) (102)

in order to obtain120572

119899(sdot) 997888rarr 119861 (sdot) weakly in 119863 [0 1] (103)

where 119861(sdot) is the zero mean Gaussian process with covariancestructure defined by

1205902(119909 119910) = 119865 (119909 and 119910) minus 119865 (119909) 119865 (119910)

+

infin

sum

119895=1

[ Cov (119868 [1198831le 119909] 119868 [119883

119895+1le 119910])

+ Cov (119868 [1198831le 119910] 119868 [119883

119895+1le 119909])]

(104)

and 119909 119910 isin [0 1]

Proof The remaining covergence of finite-dimensional dis-tributions of 120572

119899(119909) is established in [1]

ISRN Probability and Statistics 11

5 Weak Convergence of theRecursive Kernel-Type Empirical Processunder IID Assumption

The aim of this section is to show weak convergence of theempirical process

120572119899(119909) =

1

radic119899

119899

sum

119895=1

[119870(

119909 minus 119883119895

ℎ119895

) minus 119864119870(

119909 minus 119883119895

ℎ119895

)] (105)

built on iid rvrsquos 119883119895119895ge1

Wewill prove that120572

119899(sdot) convergesweakly to the Brownian

bridge 119861(sdot) with the following covariance structure

1205902(119909 119910) = 119865 (119909 and 119910) minus 119865 (119909) 119865 (119910) for 119909 119910 isin [0 1]

(106)

It turns out that the tightness criterion (8) suffices to reachthe goal Convergence of finite-dimensional distributions of120572

119899(sdot) holds and we shall show itProceeding like Cai and Roussas in [1] we need to show

that for any 119886 119887 isin R

119886120572119899(119909) + 119887120572

119899(119910) 997888rarr 119886119861 (119909) + 119887119861 (119910) in distribution

(107)

Let us introduce the notation

119884119894(119909) = 119870(

119909 minus 119883119894

ℎ119894

) minus 119864119870(119909 minus 119883

119894

ℎ119894

) (108)

and look into the covariance structure of 120572119899(sdot)

Cov (120572119899(119909) 120572

119899(119910))

= Cov( 1

radic119899

119899

sum

119894=1

119884119894(119909)

1

radic119899

119899

sum

119894=1

119884119894(119910))

=1

119899

119899

sum

119894=1

Cov (119884119894(119909) 119884

119894(119910))

=1

119899

119899

sum

119894=1

Cov(119870(119909 minus 119883

119894

ℎ119894

) 119870(119910 minus 119883

119894

ℎ119894

))

(109)

where the second equality follows from assumed indepen-dence of rvrsquos 119883

119894119894ge1

Firstly we observe that each summand converges to119865(119909and

119910) minus 119865(119909)119865(119910) since

Cov(119870(119909 minus 119883

119894

ℎ119894

) 119870(119910 minus 119883

119894

ℎ119894

))

= 119864[119870(119909 minus 119883

119894

ℎ119894

)119870(119910 minus 119883

119894

ℎ119894

)]

minus 119864119870(119909 minus 119883

119894

ℎ119894

)119864119870(119910 minus 119883

119894

ℎ119894

)

(110)

where

119864[119870(119909 minus 119883

119894

ℎ119894

)119870(119910 minus 119883

119894

ℎ119894

)]

= int

119909and119910

minusinfin

119870(119909 minus 119905

ℎ119894

)119870(119910 minus 119905

ℎ119894

)119889119865 (119905)

+ int

119909or119910

119909and119910

119870(119909 minus 119905

ℎ119894

)119870(119910 minus 119905

ℎ119894

)119889119865 (119905)

+ int

infin

119909or119910

119870(119909 minus 119905

ℎ119894

)119870(119910 minus 119905

ℎ119894

)119889119865 (119905)

(111)

and 119865 denotes the common df of the rvrsquos 119883119894119894ge1

In [6] itwas shown that

119864119870(119909 minus 119883

119894

ℎ119894

) 997888rarr 119865 (119909) (112)

and recalling that the kernel function 119870 is a df as well wearrive at the conclusion

Secondly applying Toeplitz lemma we get

Cov (120572119899(119909) 120572

119899(119910)) 997888rarr 119865 (119909 and 119910) minus 119865 (119909) 119865 (119910)

= 1205902(119909 119910) 119899 997888rarr infin

(113)

Thus

Var (119886120572119899(119909) + 119887120572

119899(119910))

= 1198862 Var (120572

119899(119909)) + 2119886119887Cov (120572

119899(119909) 120572

119899(119910))

+ 1198872 Var (120572

119899(119910))

997888rarr 1198862120590

2(119909 119909) + 2119886119887120590

2(119909 119910) + 119887

2120590

2(119910 119910) 119899 997888rarr infin

(114)

We are now on the way to prove that 119886120572119899(119909) + 119887120572

119899(119910) con-

verges in distribution to 119886119861(119909) + 119887119861(119910) sim N(0 1198862120590

2(119909 119909) +

21198861198871205902(119909 119910) + 119887

2120590

2(119910 119910)) which in other words means that

1

radic119899

infin

sum

119894=1

119886119884119894(119909) + 119887119884

119894(119910)

radicVar (119886120572119899(119909) + 119887120572

119899(119910))

997888rarr N (0 1) (115)

In order to obtain (115) we will use Lyapunov condition

lim119899rarrinfin

1

1198632+120575

119899

119899

sum

119894=1

1198641003816100381610038161003816119886119884119894

(119909) + 119887119884119894(119910)

1003816100381610038161003816

2+120575

= 0 (116)

for 120575 = 1 where 1198632

119899= Var(sum119899

119894=1[119886119884

119894(119909) + 119887119884

119894(119910)]) Since

|119870(sdot) minus 119864119870(sdot)| le 2

1198641003816100381610038161003816119886119884119894

(119909) + 119887119884i (119910)1003816100381610038161003816

3

le 8 (1198863+ 119887

3) (117)

Moreover 1198632

119899= 119899Var(119886120572

119899(119909) + 119887120572

119899(119910)) together with (114)

yields

1198633

119899= 119874 (119899

32) (118)

12 ISRN Probability and Statistics

Combining (117) with (118) we obtain

1

1198633

119899

119899

sum

119894=1

1198641003816100381610038161003816119886119884119894

(119909) + 119887119884119894(119910)

1003816100381610038161003816

3

= 119874 (119899minus12

) (119)

which shows that Lyapunov condition holds and completesthe proof of convergence of finite-dimensional distributionsof 120572

119899(119909)We summarize the result of that section in the following

theorem

Theorem 7 Let 120572119899(119909) be the recursive kernel-type empirical

process defined by (105) built on iid rvrsquos 119883119895119895ge1

withmarginal df 119865 If 120572

119899(sdot) satisfies the tightness criterion (8) then

120572119899(sdot) 997888rarr 119861 (sdot) weakly in 119863 [0 1] (120)

where 119861 is the Brownian bridge

Acknowledgment

The author would like to thank the referees for the carefulreading of the paper and helpful remarks

References

[1] Z Cai and G G Roussas ldquoWeak convergence for smoothestimator of a distribution function under negative associationrdquoStochastic Analysis and Applications vol 17 no 2 pp 145ndash1681999

[2] Z Cai andG G Roussas ldquoEfficient Estimation of a DistributionFunction under Qudrant Dependencerdquo Scandinavian Journal ofStatistics vol 25 no 1 pp 211ndash224 1998

[3] ZW Cai and G G Roussas ldquoUniform strong estimation under120572-mixing with ratesrdquo Statistics amp Probability Letters vol 15 no1 pp 47ndash55 1992

[4] G G Roussas ldquoKernel estimates under association stronguniform consistencyrdquo Statistics amp Probability Letters vol 12 no5 pp 393ndash403 1991

[5] Y F Li and S C Yang ldquoUniformly asymptotic normalityof the smooth estimation of the distribution function underassociated samplesrdquo Acta Mathematicae Applicatae Sinica vol28 no 4 pp 639ndash651 2005

[6] Y Li C Wei and S Yang ldquoThe recursive kernel distributionfunction estimator based onnegatively and positively associatedsequencesrdquo Communications in Statistics vol 39 no 20 pp3585ndash3595 2010

[7] H Yu ldquoA Glivenko-Cantelli lemma and weak convergence forempirical processes of associated sequencesrdquo ProbabilityTheoryand Related Fields vol 95 no 3 pp 357ndash370 1993

[8] Q M Shao and H Yu ldquoWeak convergence for weightedempirical processes of dependent sequencesrdquo The Annals ofProbability vol 24 no 4 pp 2098ndash2127 1996

[9] P Billingsley Convergence of Probability Measures Wiley Seriesin Probability and Statistics Probability and Statistics JohnWiley amp Sons New York NY USA 1999

[10] J D Esary F Proschan and D W Walkup ldquoAssociation ofrandom variables with applicationsrdquo Annals of MathematicalStatistics vol 38 pp 1466ndash1474 1967

[11] A Bulinski and A Shashkin Limit Theorems for AssociatedRandom Fields and Related Systems vol 10 of Advanced Serieson Statistical Science and Applied Probability World Scientific2007

[12] S Karlin and Y Rinott ldquoClasses of orderings of measures andrelated correlation inequalities I Multivariate totally positivedistributionsrdquo Journal of Multivariate Analysis vol 10 no 4 pp467ndash498 1980

[13] P Matuła and M Ziemba ldquoGeneralized covariance inequali-tiesrdquo Central European Journal of Mathematics vol 9 no 2 pp281ndash293 2011

[14] P Doukhan and S Louhichi ldquoA new weak dependence condi-tion and applications to moment inequalitiesrdquo Stochastic Proc-esses and their Applications vol 84 no 2 pp 313ndash342 1999

[15] P Matuła ldquoA note on some inequalities for certain classes ofpositively dependent random variablesrdquo Probability and Math-ematical Statistics vol 24 no 1 pp 17ndash26 2004

[16] S Louhichi ldquoWeak convergence for empirical processes of asso-ciated sequencesrdquo Annales de lrsquoInstitut Henri Poincare vol 36no 5 pp 547ndash567 2000

[17] K Joag-Dev and F Proschan ldquoNegative association of randomvariables with applicationsrdquoThe Annals of Statistics vol 11 no1 pp 286ndash295 1983

[18] C M Newman ldquoAsymptotic independence and limit theoremsfor positively and negatively dependent random variablesrdquo inInequalities in Statistics and Probability Y L Tong Ed vol 5pp 127ndash140 Institute ofMathematical StatisticsHayward CalifUSA 1984

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

ISRN Probability and Statistics 9

It turns out that it suffices to have

1003816100381610038161003816Cov (1198801 119880

1+119899)1003816100381610038161003816 = 119874(

1

1198993119901(119901minus1)) (82)

where 119901 gt 4 is a positive constant taken from the tightnesscriterion (8) It is easy to see that asymptotically we get therate 3

On the way to prove it we will also take use of aRosenthal-type inequality due to Shao andYu (seeTheorem 2in [8]) we shall recall in the following lemma

Lemma 5 Let 119901 gt 2 and 119891 be a real valued function boundedby 1 with bounded first derivative Suppose that 119883

119899119899ge1

is asequence of stationary and associated rvrsquos such that for 119899 isin N

Cov (1198831 119883

119899) = 119874 (119899

minus119887) for some 119887 gt 119901 minus 1 (83)

Then for any 120583 gt 0 there exists some positive constant 119896120583

independent of the function 119891 for which

1198641003816100381610038161003816119878119899

(119891) minus 119864119878119899(119891)

1003816100381610038161003816

119901

le 119896120583(119899

1+12058310038171003817100381710038171003817119891

101584010038171003817100381710038171003817

2

+1198991199012 [

[

119899

sum

119895=1

10038161003816100381610038161003816Cov (119891 (119883

1)119891 (119883

119895))10038161003816100381610038161003816]

]

)

(84)

As we can see the lemma assumes association but itworks for negatively associated rvrsquos as well since in the proofit reaches back the result of Newman (see Proposition 15 in[18]) where both types of association are allowed

Let us recall that 120572119899(119909) = radic119899(119865

119899(119909) minus 119864119865

119899(119909)) where

119865119899(119909) = (1119899)sum

119899

119895=1119870((119909 minus 119883

119895)ℎ

119899) It is easy to see that

1198641003816100381610038161003816120572119899

(119909) minus 120572119899(119910)

1003816100381610038161003816

119901

= 119899minus1199012

11986410038161003816100381610038161003816119878119899(119891 (119883

119895)) minus 119864119878

119899(119891 (119883

119895))10038161003816100381610038161003816

119901

(85)

where

119891 (119883119895) = 119870(

119909 minus 119883119895

ℎ119899

) minus 119870(

119910 minus 119883119895

ℎ119899

)

119878119899(119891) =

119899

sum

119895=1

119891 (119883119895)

(86)

With an intent to use Lemma 5 we need 119891 to be boundedfrom above by 1 (which in our case is obvious) and to havebounded 119891

1015840 thus we assume

sup119905isinR

10038161003816100381610038161003816119870

1015840(119905)

10038161003816100381610038161003816

1

ℎ119899

le119862

119870

ℎ119899

lt infin (87)

Now applying inequality (84) we have

1198641003816100381610038161003816120572119899

(119909) minus 120572119899(119910)

1003816100381610038161003816

119901

le119896120583119899minus1199012

1198991+120583

1198622

119870

ℎ2

119899

+1198991199012[

[

119899

sum

119895=1

10038161003816100381610038161003816Cov (119891 (119883

1)119891 (119883

119895))10038161003816100381610038161003816]

]

1199012

(88)

Newman showed in [18] that

Cov (1198911(119883) 119891

2(119884))

= ∬R2119891

1015840

1(119909) 119891

1015840

2(119910)

times [119875 (119883 le 119909 119884 le 119910) minus 119875 (119883 le 119909) 119875 (119884 le 119910)] 119889119909 119889119910

(89)

if 1198911and 119891

2are real valued functions on R having square

integrable derivatives 1198911015840

1and 119891

1015840

2 respectively and provided

that 1198911(119883) 119891

2(119884) have finite secondmoments In light of that

equation and linearity of covariance we get10038161003816100381610038161003816Cov (119891 (119883

1) 119891 (119883

119895))10038161003816100381610038161003816

=

1003816100381610038161003816100381610038161003816∬

R2[119875 (119883

1lt 119909 minus ℎ

119899119904 119883

119895lt 119909 minus ℎ

119899119905)

minus119875 (1198831lt 119909 minus ℎ

119899119904 119883

119895lt 119910 minus ℎ

119899119905)]

minus [119875 (1198831lt 119909 minus ℎ

119899119904) 119875 (119883

119895lt 119909 minus ℎ

119899119905)

minus119875 (1198831lt 119909 minus ℎ

119899119904) 119875 (119883

119895lt 119910 minus ℎ

119899119905)]

minus [119875 (1198831lt 119910 minus ℎ

119899119904 119883

119895lt 119909 minus ℎ

119899119905)

minus119875 (1198831lt 119910 minus ℎ

119899119904 119883

119895lt 119910 minus ℎ

119899119905)]

minus [119875 (1198831lt 119910 minus ℎ

119899119904) 119875 (119883

119895lt 119910 minus ℎ

119899119905)

minus 119875 (1198831lt 119910 minus ℎ

119899119904)

times 119875 (119883119895lt 119909 minus ℎ

119899119905)] 119896 (119904) 119896 (119905) 119889119904 119889119905

10038161003816100381610038161003816100381610038161003816

(90)

Without the loss of generality we may and do assume that119909 gt 119910 thus10038161003816100381610038161003816Cov (119891 (119883

1) 119891 (119883

119895))10038161003816100381610038161003816

=

1003816100381610038161003816100381610038161003816∬

R2119875 (119910 minus ℎ

119899119904 le 119883

1lt 119909 minus ℎ

119899119904

119910 minus ℎ119899119905 le 119883

119895lt 119909 minus ℎ

119899119905)

minus 119875 (119910 minus ℎ119899119904 le 119883

1lt 119909 minus ℎ

119899119904)

times 119875 (119910 minus ℎ119899119905 le 119883

119895lt 119909 minus ℎ

119899119905) 119896 (119904) 119896 (119905) 119889119904 119889119905

1003816100381610038161003816100381610038161003816

(91)

and by triangle inequality we get10038161003816100381610038161003816Cov (119891 (119883

1) 119891 (119883

119895))10038161003816100381610038161003816

le ∬R2[int

119909minusℎ119899119904

119910minusℎ119899119904

int

119909minusℎ119899119905

119910minusℎ119899119905

1198911198831119883119895(119906 V) 119889119906 119889V

+int

119909minusℎ119899119904

119910minusℎ119899119904

1198911198831(119906) 119889119906int

119909minusℎ119899119905

119910minusℎ119899119905

119891119883119895(V) 119889V] 119896 (119904) 119896 (119905) 119889119904 119889119905

(92)

10 ISRN Probability and Statistics

1198911198831119883119895

(119906 V) 1198911198831

(119906) and 119891119883119895

(V) are the joint pdf of [1198831 119883

119895]

and marginal pdfrsquos of rvrsquos1198831and119883

119895 respectively We need

to further assume that 1198621stands for a common upper bound

of 1198911198831

and 1198911198831119883119895

for all 119895 isin N that is

100381710038171003817100381710038171198911198831

10038171003817100381710038171003817le 119862

1

1003817100381710038171003817100381710038171198911198831119883119895

100381710038171003817100381710038171003817le 119862

1forall119895 ge 1 (93)

Then we finally obtain

10038161003816100381610038161003816Cov (119891 (119883

1) 119891 (119883

119895))10038161003816100381610038161003816le 2119862

1

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

2

where = max 1198621 119862

2

1

(94)

On the other hand proceeding like Cai and Roussas in [1] wecan shortly get

10038161003816100381610038161003816Cov (119891 (119883

1) 119891 (119883

119895))10038161003816100381610038161003816

le

100381610038161003816100381610038161003816100381610038161003816

Cov(119870(119909 minus 119883

1

ℎ119899

) 119870(

119909 minus 119883119895

ℎ119899

))

100381610038161003816100381610038161003816100381610038161003816

+

100381610038161003816100381610038161003816100381610038161003816

Cov(119870(119909 minus 119883

1

ℎ119899

) 119870(

119910 minus 119883119895

ℎ119899

))

100381610038161003816100381610038161003816100381610038161003816

+

100381610038161003816100381610038161003816100381610038161003816

Cov(119870(119910 minus 119883

1

ℎ119899

) 119870(

119909 minus 119883119895

ℎ119899

))

100381610038161003816100381610038161003816100381610038161003816

+

100381610038161003816100381610038161003816100381610038161003816

Cov(119870(119910 minus 119883

1

ℎ119899

) 119870(

119910 minus 119883119895

ℎ119899

))

100381610038161003816100381610038161003816100381610038161003816

le 41198622

10038161003816100381610038161003816Cov (119883

1 119883

119895)10038161003816100381610038161003816

13

(95)

where 1198622is a constant relevant to the covariance inequality

for negatively associated rvrsquos (see [13])We now arrive at the following inequality

10038161003816100381610038161003816Cov (119891 (119883

1) 119891 (119883

119895))10038161003816100381610038161003816

le max 1198621 119862

2

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

2

10038161003816100381610038161003816Cov(119883

1 119883

119895)10038161003816100381610038161003816

13

(96)

Assuming |Cov(119891(1198831) 119891(119883

119895))| = 119874(119895

minus119903) and proceeding

similarly to (68) we can get

[

[

119899

sum

119895=1

10038161003816100381610038161003816Cov (119891 (119883

1) 119891 (119883

119895))10038161003816100381610038161003816]

]

1199012

le max 1198621 119862

2 [

[

sum

119895lt|119909minus119910|minus2(1199033)

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

2

+ sum

119895ge|119909minus119910|minus2(1199033)

1

1198951199033]

]

1199012

le 2max 1198621 119862

21003816100381610038161003816119909 minus 119910

1003816100381610038161003816

((119903minus3)119903)119901

(97)

To sum up we obtain the following inequality

1198641003816100381610038161003816120572119899

(119909) minus 120572119899(119910)

1003816100381610038161003816

119901

le 119896120583119899

1+120583minus1199012119862

2

119870

ℎ2

119899

+ 2max 1198621 119862

21003816100381610038161003816119909 minus 119910

1003816100381610038161003816

((119903minus3)119903)119901

le 1198621198991+120583minus1199012 1

ℎ2

119899

+1003816100381610038161003816119909 minus 119910

1003816100381610038161003816

((119903minus3)119903)119901

(98)

where 119862 = 119896120583max1198622

119896 2max119862

1 119862

2 The formula of the

tightness criterion (8) implies that ((119903 minus 3)119903)119901 gt 1 so

119903 gt3119901

119901 minus 1 where 119901 gt 2 (99)

Let us recall the assumptions imposed on the bandwidhtsℎ

119899119899ge1

by Cai and Roussas in [1]B1 lim

119899rarrinfinℎ119899= 0 and ℎ

119899gt 0 for all 119899 isin N

+

B2 lim119899rarrinfin

119899ℎ119899= infin thus ℎ

119899= 119874(1119899

1minus120573) 120573 gt 0

B3 lim119899rarrinfin

119899ℎ4

119899= 0 hence ℎ

119899= 119874(1119899

14+120575) 120575 gt 0

In light of the above

1198991+120583minus1199012 1

ℎ2

119899

= 11989932+120583+2120575minus1199012

(100)

where 120583 gt 0 119901 gt 2 and 0 lt 120575 lt 34 Confrontation with thetightness criterion (8) forces

3

2+ 120583 + 2120575 minus

119901

2lt minus

1

2 (101)

which implies 119901 gt 4 Let us conclude with the followingtheorem

Theorem6 Let 120572119899(119909) = radic119899(119865

119899(119909)minus119864119865

119899(119909)) be the empirical

process built on the kernel estimator of the df119865 for a stationarysequence of negatively associated rvrsquos Assume that conditionsA1 A2 (87) B1 B2 B3 are satisfied Then provided that thetightness criterion (8) holds with 119901 gt 4 it suffices to demand

1003816100381610038161003816Cov (1198801 119880

1+119899)1003816100381610038161003816 = 119874(

1

1198993119901(119901minus1)) (102)

in order to obtain120572

119899(sdot) 997888rarr 119861 (sdot) weakly in 119863 [0 1] (103)

where 119861(sdot) is the zero mean Gaussian process with covariancestructure defined by

1205902(119909 119910) = 119865 (119909 and 119910) minus 119865 (119909) 119865 (119910)

+

infin

sum

119895=1

[ Cov (119868 [1198831le 119909] 119868 [119883

119895+1le 119910])

+ Cov (119868 [1198831le 119910] 119868 [119883

119895+1le 119909])]

(104)

and 119909 119910 isin [0 1]

Proof The remaining covergence of finite-dimensional dis-tributions of 120572

119899(119909) is established in [1]

ISRN Probability and Statistics 11

5 Weak Convergence of theRecursive Kernel-Type Empirical Processunder IID Assumption

The aim of this section is to show weak convergence of theempirical process

120572119899(119909) =

1

radic119899

119899

sum

119895=1

[119870(

119909 minus 119883119895

ℎ119895

) minus 119864119870(

119909 minus 119883119895

ℎ119895

)] (105)

built on iid rvrsquos 119883119895119895ge1

Wewill prove that120572

119899(sdot) convergesweakly to the Brownian

bridge 119861(sdot) with the following covariance structure

1205902(119909 119910) = 119865 (119909 and 119910) minus 119865 (119909) 119865 (119910) for 119909 119910 isin [0 1]

(106)

It turns out that the tightness criterion (8) suffices to reachthe goal Convergence of finite-dimensional distributions of120572

119899(sdot) holds and we shall show itProceeding like Cai and Roussas in [1] we need to show

that for any 119886 119887 isin R

119886120572119899(119909) + 119887120572

119899(119910) 997888rarr 119886119861 (119909) + 119887119861 (119910) in distribution

(107)

Let us introduce the notation

119884119894(119909) = 119870(

119909 minus 119883119894

ℎ119894

) minus 119864119870(119909 minus 119883

119894

ℎ119894

) (108)

and look into the covariance structure of 120572119899(sdot)

Cov (120572119899(119909) 120572

119899(119910))

= Cov( 1

radic119899

119899

sum

119894=1

119884119894(119909)

1

radic119899

119899

sum

119894=1

119884119894(119910))

=1

119899

119899

sum

119894=1

Cov (119884119894(119909) 119884

119894(119910))

=1

119899

119899

sum

119894=1

Cov(119870(119909 minus 119883

119894

ℎ119894

) 119870(119910 minus 119883

119894

ℎ119894

))

(109)

where the second equality follows from assumed indepen-dence of rvrsquos 119883

119894119894ge1

Firstly we observe that each summand converges to119865(119909and

119910) minus 119865(119909)119865(119910) since

Cov(119870(119909 minus 119883

119894

ℎ119894

) 119870(119910 minus 119883

119894

ℎ119894

))

= 119864[119870(119909 minus 119883

119894

ℎ119894

)119870(119910 minus 119883

119894

ℎ119894

)]

minus 119864119870(119909 minus 119883

119894

ℎ119894

)119864119870(119910 minus 119883

119894

ℎ119894

)

(110)

where

119864[119870(119909 minus 119883

119894

ℎ119894

)119870(119910 minus 119883

119894

ℎ119894

)]

= int

119909and119910

minusinfin

119870(119909 minus 119905

ℎ119894

)119870(119910 minus 119905

ℎ119894

)119889119865 (119905)

+ int

119909or119910

119909and119910

119870(119909 minus 119905

ℎ119894

)119870(119910 minus 119905

ℎ119894

)119889119865 (119905)

+ int

infin

119909or119910

119870(119909 minus 119905

ℎ119894

)119870(119910 minus 119905

ℎ119894

)119889119865 (119905)

(111)

and 119865 denotes the common df of the rvrsquos 119883119894119894ge1

In [6] itwas shown that

119864119870(119909 minus 119883

119894

ℎ119894

) 997888rarr 119865 (119909) (112)

and recalling that the kernel function 119870 is a df as well wearrive at the conclusion

Secondly applying Toeplitz lemma we get

Cov (120572119899(119909) 120572

119899(119910)) 997888rarr 119865 (119909 and 119910) minus 119865 (119909) 119865 (119910)

= 1205902(119909 119910) 119899 997888rarr infin

(113)

Thus

Var (119886120572119899(119909) + 119887120572

119899(119910))

= 1198862 Var (120572

119899(119909)) + 2119886119887Cov (120572

119899(119909) 120572

119899(119910))

+ 1198872 Var (120572

119899(119910))

997888rarr 1198862120590

2(119909 119909) + 2119886119887120590

2(119909 119910) + 119887

2120590

2(119910 119910) 119899 997888rarr infin

(114)

We are now on the way to prove that 119886120572119899(119909) + 119887120572

119899(119910) con-

verges in distribution to 119886119861(119909) + 119887119861(119910) sim N(0 1198862120590

2(119909 119909) +

21198861198871205902(119909 119910) + 119887

2120590

2(119910 119910)) which in other words means that

1

radic119899

infin

sum

119894=1

119886119884119894(119909) + 119887119884

119894(119910)

radicVar (119886120572119899(119909) + 119887120572

119899(119910))

997888rarr N (0 1) (115)

In order to obtain (115) we will use Lyapunov condition

lim119899rarrinfin

1

1198632+120575

119899

119899

sum

119894=1

1198641003816100381610038161003816119886119884119894

(119909) + 119887119884119894(119910)

1003816100381610038161003816

2+120575

= 0 (116)

for 120575 = 1 where 1198632

119899= Var(sum119899

119894=1[119886119884

119894(119909) + 119887119884

119894(119910)]) Since

|119870(sdot) minus 119864119870(sdot)| le 2

1198641003816100381610038161003816119886119884119894

(119909) + 119887119884i (119910)1003816100381610038161003816

3

le 8 (1198863+ 119887

3) (117)

Moreover 1198632

119899= 119899Var(119886120572

119899(119909) + 119887120572

119899(119910)) together with (114)

yields

1198633

119899= 119874 (119899

32) (118)

12 ISRN Probability and Statistics

Combining (117) with (118) we obtain

1

1198633

119899

119899

sum

119894=1

1198641003816100381610038161003816119886119884119894

(119909) + 119887119884119894(119910)

1003816100381610038161003816

3

= 119874 (119899minus12

) (119)

which shows that Lyapunov condition holds and completesthe proof of convergence of finite-dimensional distributionsof 120572

119899(119909)We summarize the result of that section in the following

theorem

Theorem 7 Let 120572119899(119909) be the recursive kernel-type empirical

process defined by (105) built on iid rvrsquos 119883119895119895ge1

withmarginal df 119865 If 120572

119899(sdot) satisfies the tightness criterion (8) then

120572119899(sdot) 997888rarr 119861 (sdot) weakly in 119863 [0 1] (120)

where 119861 is the Brownian bridge

Acknowledgment

The author would like to thank the referees for the carefulreading of the paper and helpful remarks

References

[1] Z Cai and G G Roussas ldquoWeak convergence for smoothestimator of a distribution function under negative associationrdquoStochastic Analysis and Applications vol 17 no 2 pp 145ndash1681999

[2] Z Cai andG G Roussas ldquoEfficient Estimation of a DistributionFunction under Qudrant Dependencerdquo Scandinavian Journal ofStatistics vol 25 no 1 pp 211ndash224 1998

[3] ZW Cai and G G Roussas ldquoUniform strong estimation under120572-mixing with ratesrdquo Statistics amp Probability Letters vol 15 no1 pp 47ndash55 1992

[4] G G Roussas ldquoKernel estimates under association stronguniform consistencyrdquo Statistics amp Probability Letters vol 12 no5 pp 393ndash403 1991

[5] Y F Li and S C Yang ldquoUniformly asymptotic normalityof the smooth estimation of the distribution function underassociated samplesrdquo Acta Mathematicae Applicatae Sinica vol28 no 4 pp 639ndash651 2005

[6] Y Li C Wei and S Yang ldquoThe recursive kernel distributionfunction estimator based onnegatively and positively associatedsequencesrdquo Communications in Statistics vol 39 no 20 pp3585ndash3595 2010

[7] H Yu ldquoA Glivenko-Cantelli lemma and weak convergence forempirical processes of associated sequencesrdquo ProbabilityTheoryand Related Fields vol 95 no 3 pp 357ndash370 1993

[8] Q M Shao and H Yu ldquoWeak convergence for weightedempirical processes of dependent sequencesrdquo The Annals ofProbability vol 24 no 4 pp 2098ndash2127 1996

[9] P Billingsley Convergence of Probability Measures Wiley Seriesin Probability and Statistics Probability and Statistics JohnWiley amp Sons New York NY USA 1999

[10] J D Esary F Proschan and D W Walkup ldquoAssociation ofrandom variables with applicationsrdquo Annals of MathematicalStatistics vol 38 pp 1466ndash1474 1967

[11] A Bulinski and A Shashkin Limit Theorems for AssociatedRandom Fields and Related Systems vol 10 of Advanced Serieson Statistical Science and Applied Probability World Scientific2007

[12] S Karlin and Y Rinott ldquoClasses of orderings of measures andrelated correlation inequalities I Multivariate totally positivedistributionsrdquo Journal of Multivariate Analysis vol 10 no 4 pp467ndash498 1980

[13] P Matuła and M Ziemba ldquoGeneralized covariance inequali-tiesrdquo Central European Journal of Mathematics vol 9 no 2 pp281ndash293 2011

[14] P Doukhan and S Louhichi ldquoA new weak dependence condi-tion and applications to moment inequalitiesrdquo Stochastic Proc-esses and their Applications vol 84 no 2 pp 313ndash342 1999

[15] P Matuła ldquoA note on some inequalities for certain classes ofpositively dependent random variablesrdquo Probability and Math-ematical Statistics vol 24 no 1 pp 17ndash26 2004

[16] S Louhichi ldquoWeak convergence for empirical processes of asso-ciated sequencesrdquo Annales de lrsquoInstitut Henri Poincare vol 36no 5 pp 547ndash567 2000

[17] K Joag-Dev and F Proschan ldquoNegative association of randomvariables with applicationsrdquoThe Annals of Statistics vol 11 no1 pp 286ndash295 1983

[18] C M Newman ldquoAsymptotic independence and limit theoremsfor positively and negatively dependent random variablesrdquo inInequalities in Statistics and Probability Y L Tong Ed vol 5pp 127ndash140 Institute ofMathematical StatisticsHayward CalifUSA 1984

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 ISRN Probability and Statistics

1198911198831119883119895

(119906 V) 1198911198831

(119906) and 119891119883119895

(V) are the joint pdf of [1198831 119883

119895]

and marginal pdfrsquos of rvrsquos1198831and119883

119895 respectively We need

to further assume that 1198621stands for a common upper bound

of 1198911198831

and 1198911198831119883119895

for all 119895 isin N that is

100381710038171003817100381710038171198911198831

10038171003817100381710038171003817le 119862

1

1003817100381710038171003817100381710038171198911198831119883119895

100381710038171003817100381710038171003817le 119862

1forall119895 ge 1 (93)

Then we finally obtain

10038161003816100381610038161003816Cov (119891 (119883

1) 119891 (119883

119895))10038161003816100381610038161003816le 2119862

1

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

2

where = max 1198621 119862

2

1

(94)

On the other hand proceeding like Cai and Roussas in [1] wecan shortly get

10038161003816100381610038161003816Cov (119891 (119883

1) 119891 (119883

119895))10038161003816100381610038161003816

le

100381610038161003816100381610038161003816100381610038161003816

Cov(119870(119909 minus 119883

1

ℎ119899

) 119870(

119909 minus 119883119895

ℎ119899

))

100381610038161003816100381610038161003816100381610038161003816

+

100381610038161003816100381610038161003816100381610038161003816

Cov(119870(119909 minus 119883

1

ℎ119899

) 119870(

119910 minus 119883119895

ℎ119899

))

100381610038161003816100381610038161003816100381610038161003816

+

100381610038161003816100381610038161003816100381610038161003816

Cov(119870(119910 minus 119883

1

ℎ119899

) 119870(

119909 minus 119883119895

ℎ119899

))

100381610038161003816100381610038161003816100381610038161003816

+

100381610038161003816100381610038161003816100381610038161003816

Cov(119870(119910 minus 119883

1

ℎ119899

) 119870(

119910 minus 119883119895

ℎ119899

))

100381610038161003816100381610038161003816100381610038161003816

le 41198622

10038161003816100381610038161003816Cov (119883

1 119883

119895)10038161003816100381610038161003816

13

(95)

where 1198622is a constant relevant to the covariance inequality

for negatively associated rvrsquos (see [13])We now arrive at the following inequality

10038161003816100381610038161003816Cov (119891 (119883

1) 119891 (119883

119895))10038161003816100381610038161003816

le max 1198621 119862

2

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

2

10038161003816100381610038161003816Cov(119883

1 119883

119895)10038161003816100381610038161003816

13

(96)

Assuming |Cov(119891(1198831) 119891(119883

119895))| = 119874(119895

minus119903) and proceeding

similarly to (68) we can get

[

[

119899

sum

119895=1

10038161003816100381610038161003816Cov (119891 (119883

1) 119891 (119883

119895))10038161003816100381610038161003816]

]

1199012

le max 1198621 119862

2 [

[

sum

119895lt|119909minus119910|minus2(1199033)

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

2

+ sum

119895ge|119909minus119910|minus2(1199033)

1

1198951199033]

]

1199012

le 2max 1198621 119862

21003816100381610038161003816119909 minus 119910

1003816100381610038161003816

((119903minus3)119903)119901

(97)

To sum up we obtain the following inequality

1198641003816100381610038161003816120572119899

(119909) minus 120572119899(119910)

1003816100381610038161003816

119901

le 119896120583119899

1+120583minus1199012119862

2

119870

ℎ2

119899

+ 2max 1198621 119862

21003816100381610038161003816119909 minus 119910

1003816100381610038161003816

((119903minus3)119903)119901

le 1198621198991+120583minus1199012 1

ℎ2

119899

+1003816100381610038161003816119909 minus 119910

1003816100381610038161003816

((119903minus3)119903)119901

(98)

where 119862 = 119896120583max1198622

119896 2max119862

1 119862

2 The formula of the

tightness criterion (8) implies that ((119903 minus 3)119903)119901 gt 1 so

119903 gt3119901

119901 minus 1 where 119901 gt 2 (99)

Let us recall the assumptions imposed on the bandwidhtsℎ

119899119899ge1

by Cai and Roussas in [1]B1 lim

119899rarrinfinℎ119899= 0 and ℎ

119899gt 0 for all 119899 isin N

+

B2 lim119899rarrinfin

119899ℎ119899= infin thus ℎ

119899= 119874(1119899

1minus120573) 120573 gt 0

B3 lim119899rarrinfin

119899ℎ4

119899= 0 hence ℎ

119899= 119874(1119899

14+120575) 120575 gt 0

In light of the above

1198991+120583minus1199012 1

ℎ2

119899

= 11989932+120583+2120575minus1199012

(100)

where 120583 gt 0 119901 gt 2 and 0 lt 120575 lt 34 Confrontation with thetightness criterion (8) forces

3

2+ 120583 + 2120575 minus

119901

2lt minus

1

2 (101)

which implies 119901 gt 4 Let us conclude with the followingtheorem

Theorem6 Let 120572119899(119909) = radic119899(119865

119899(119909)minus119864119865

119899(119909)) be the empirical

process built on the kernel estimator of the df119865 for a stationarysequence of negatively associated rvrsquos Assume that conditionsA1 A2 (87) B1 B2 B3 are satisfied Then provided that thetightness criterion (8) holds with 119901 gt 4 it suffices to demand

1003816100381610038161003816Cov (1198801 119880

1+119899)1003816100381610038161003816 = 119874(

1

1198993119901(119901minus1)) (102)

in order to obtain120572

119899(sdot) 997888rarr 119861 (sdot) weakly in 119863 [0 1] (103)

where 119861(sdot) is the zero mean Gaussian process with covariancestructure defined by

1205902(119909 119910) = 119865 (119909 and 119910) minus 119865 (119909) 119865 (119910)

+

infin

sum

119895=1

[ Cov (119868 [1198831le 119909] 119868 [119883

119895+1le 119910])

+ Cov (119868 [1198831le 119910] 119868 [119883

119895+1le 119909])]

(104)

and 119909 119910 isin [0 1]

Proof The remaining covergence of finite-dimensional dis-tributions of 120572

119899(119909) is established in [1]

ISRN Probability and Statistics 11

5 Weak Convergence of theRecursive Kernel-Type Empirical Processunder IID Assumption

The aim of this section is to show weak convergence of theempirical process

120572119899(119909) =

1

radic119899

119899

sum

119895=1

[119870(

119909 minus 119883119895

ℎ119895

) minus 119864119870(

119909 minus 119883119895

ℎ119895

)] (105)

built on iid rvrsquos 119883119895119895ge1

Wewill prove that120572

119899(sdot) convergesweakly to the Brownian

bridge 119861(sdot) with the following covariance structure

1205902(119909 119910) = 119865 (119909 and 119910) minus 119865 (119909) 119865 (119910) for 119909 119910 isin [0 1]

(106)

It turns out that the tightness criterion (8) suffices to reachthe goal Convergence of finite-dimensional distributions of120572

119899(sdot) holds and we shall show itProceeding like Cai and Roussas in [1] we need to show

that for any 119886 119887 isin R

119886120572119899(119909) + 119887120572

119899(119910) 997888rarr 119886119861 (119909) + 119887119861 (119910) in distribution

(107)

Let us introduce the notation

119884119894(119909) = 119870(

119909 minus 119883119894

ℎ119894

) minus 119864119870(119909 minus 119883

119894

ℎ119894

) (108)

and look into the covariance structure of 120572119899(sdot)

Cov (120572119899(119909) 120572

119899(119910))

= Cov( 1

radic119899

119899

sum

119894=1

119884119894(119909)

1

radic119899

119899

sum

119894=1

119884119894(119910))

=1

119899

119899

sum

119894=1

Cov (119884119894(119909) 119884

119894(119910))

=1

119899

119899

sum

119894=1

Cov(119870(119909 minus 119883

119894

ℎ119894

) 119870(119910 minus 119883

119894

ℎ119894

))

(109)

where the second equality follows from assumed indepen-dence of rvrsquos 119883

119894119894ge1

Firstly we observe that each summand converges to119865(119909and

119910) minus 119865(119909)119865(119910) since

Cov(119870(119909 minus 119883

119894

ℎ119894

) 119870(119910 minus 119883

119894

ℎ119894

))

= 119864[119870(119909 minus 119883

119894

ℎ119894

)119870(119910 minus 119883

119894

ℎ119894

)]

minus 119864119870(119909 minus 119883

119894

ℎ119894

)119864119870(119910 minus 119883

119894

ℎ119894

)

(110)

where

119864[119870(119909 minus 119883

119894

ℎ119894

)119870(119910 minus 119883

119894

ℎ119894

)]

= int

119909and119910

minusinfin

119870(119909 minus 119905

ℎ119894

)119870(119910 minus 119905

ℎ119894

)119889119865 (119905)

+ int

119909or119910

119909and119910

119870(119909 minus 119905

ℎ119894

)119870(119910 minus 119905

ℎ119894

)119889119865 (119905)

+ int

infin

119909or119910

119870(119909 minus 119905

ℎ119894

)119870(119910 minus 119905

ℎ119894

)119889119865 (119905)

(111)

and 119865 denotes the common df of the rvrsquos 119883119894119894ge1

In [6] itwas shown that

119864119870(119909 minus 119883

119894

ℎ119894

) 997888rarr 119865 (119909) (112)

and recalling that the kernel function 119870 is a df as well wearrive at the conclusion

Secondly applying Toeplitz lemma we get

Cov (120572119899(119909) 120572

119899(119910)) 997888rarr 119865 (119909 and 119910) minus 119865 (119909) 119865 (119910)

= 1205902(119909 119910) 119899 997888rarr infin

(113)

Thus

Var (119886120572119899(119909) + 119887120572

119899(119910))

= 1198862 Var (120572

119899(119909)) + 2119886119887Cov (120572

119899(119909) 120572

119899(119910))

+ 1198872 Var (120572

119899(119910))

997888rarr 1198862120590

2(119909 119909) + 2119886119887120590

2(119909 119910) + 119887

2120590

2(119910 119910) 119899 997888rarr infin

(114)

We are now on the way to prove that 119886120572119899(119909) + 119887120572

119899(119910) con-

verges in distribution to 119886119861(119909) + 119887119861(119910) sim N(0 1198862120590

2(119909 119909) +

21198861198871205902(119909 119910) + 119887

2120590

2(119910 119910)) which in other words means that

1

radic119899

infin

sum

119894=1

119886119884119894(119909) + 119887119884

119894(119910)

radicVar (119886120572119899(119909) + 119887120572

119899(119910))

997888rarr N (0 1) (115)

In order to obtain (115) we will use Lyapunov condition

lim119899rarrinfin

1

1198632+120575

119899

119899

sum

119894=1

1198641003816100381610038161003816119886119884119894

(119909) + 119887119884119894(119910)

1003816100381610038161003816

2+120575

= 0 (116)

for 120575 = 1 where 1198632

119899= Var(sum119899

119894=1[119886119884

119894(119909) + 119887119884

119894(119910)]) Since

|119870(sdot) minus 119864119870(sdot)| le 2

1198641003816100381610038161003816119886119884119894

(119909) + 119887119884i (119910)1003816100381610038161003816

3

le 8 (1198863+ 119887

3) (117)

Moreover 1198632

119899= 119899Var(119886120572

119899(119909) + 119887120572

119899(119910)) together with (114)

yields

1198633

119899= 119874 (119899

32) (118)

12 ISRN Probability and Statistics

Combining (117) with (118) we obtain

1

1198633

119899

119899

sum

119894=1

1198641003816100381610038161003816119886119884119894

(119909) + 119887119884119894(119910)

1003816100381610038161003816

3

= 119874 (119899minus12

) (119)

which shows that Lyapunov condition holds and completesthe proof of convergence of finite-dimensional distributionsof 120572

119899(119909)We summarize the result of that section in the following

theorem

Theorem 7 Let 120572119899(119909) be the recursive kernel-type empirical

process defined by (105) built on iid rvrsquos 119883119895119895ge1

withmarginal df 119865 If 120572

119899(sdot) satisfies the tightness criterion (8) then

120572119899(sdot) 997888rarr 119861 (sdot) weakly in 119863 [0 1] (120)

where 119861 is the Brownian bridge

Acknowledgment

The author would like to thank the referees for the carefulreading of the paper and helpful remarks

References

[1] Z Cai and G G Roussas ldquoWeak convergence for smoothestimator of a distribution function under negative associationrdquoStochastic Analysis and Applications vol 17 no 2 pp 145ndash1681999

[2] Z Cai andG G Roussas ldquoEfficient Estimation of a DistributionFunction under Qudrant Dependencerdquo Scandinavian Journal ofStatistics vol 25 no 1 pp 211ndash224 1998

[3] ZW Cai and G G Roussas ldquoUniform strong estimation under120572-mixing with ratesrdquo Statistics amp Probability Letters vol 15 no1 pp 47ndash55 1992

[4] G G Roussas ldquoKernel estimates under association stronguniform consistencyrdquo Statistics amp Probability Letters vol 12 no5 pp 393ndash403 1991

[5] Y F Li and S C Yang ldquoUniformly asymptotic normalityof the smooth estimation of the distribution function underassociated samplesrdquo Acta Mathematicae Applicatae Sinica vol28 no 4 pp 639ndash651 2005

[6] Y Li C Wei and S Yang ldquoThe recursive kernel distributionfunction estimator based onnegatively and positively associatedsequencesrdquo Communications in Statistics vol 39 no 20 pp3585ndash3595 2010

[7] H Yu ldquoA Glivenko-Cantelli lemma and weak convergence forempirical processes of associated sequencesrdquo ProbabilityTheoryand Related Fields vol 95 no 3 pp 357ndash370 1993

[8] Q M Shao and H Yu ldquoWeak convergence for weightedempirical processes of dependent sequencesrdquo The Annals ofProbability vol 24 no 4 pp 2098ndash2127 1996

[9] P Billingsley Convergence of Probability Measures Wiley Seriesin Probability and Statistics Probability and Statistics JohnWiley amp Sons New York NY USA 1999

[10] J D Esary F Proschan and D W Walkup ldquoAssociation ofrandom variables with applicationsrdquo Annals of MathematicalStatistics vol 38 pp 1466ndash1474 1967

[11] A Bulinski and A Shashkin Limit Theorems for AssociatedRandom Fields and Related Systems vol 10 of Advanced Serieson Statistical Science and Applied Probability World Scientific2007

[12] S Karlin and Y Rinott ldquoClasses of orderings of measures andrelated correlation inequalities I Multivariate totally positivedistributionsrdquo Journal of Multivariate Analysis vol 10 no 4 pp467ndash498 1980

[13] P Matuła and M Ziemba ldquoGeneralized covariance inequali-tiesrdquo Central European Journal of Mathematics vol 9 no 2 pp281ndash293 2011

[14] P Doukhan and S Louhichi ldquoA new weak dependence condi-tion and applications to moment inequalitiesrdquo Stochastic Proc-esses and their Applications vol 84 no 2 pp 313ndash342 1999

[15] P Matuła ldquoA note on some inequalities for certain classes ofpositively dependent random variablesrdquo Probability and Math-ematical Statistics vol 24 no 1 pp 17ndash26 2004

[16] S Louhichi ldquoWeak convergence for empirical processes of asso-ciated sequencesrdquo Annales de lrsquoInstitut Henri Poincare vol 36no 5 pp 547ndash567 2000

[17] K Joag-Dev and F Proschan ldquoNegative association of randomvariables with applicationsrdquoThe Annals of Statistics vol 11 no1 pp 286ndash295 1983

[18] C M Newman ldquoAsymptotic independence and limit theoremsfor positively and negatively dependent random variablesrdquo inInequalities in Statistics and Probability Y L Tong Ed vol 5pp 127ndash140 Institute ofMathematical StatisticsHayward CalifUSA 1984

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

ISRN Probability and Statistics 11

5 Weak Convergence of theRecursive Kernel-Type Empirical Processunder IID Assumption

The aim of this section is to show weak convergence of theempirical process

120572119899(119909) =

1

radic119899

119899

sum

119895=1

[119870(

119909 minus 119883119895

ℎ119895

) minus 119864119870(

119909 minus 119883119895

ℎ119895

)] (105)

built on iid rvrsquos 119883119895119895ge1

Wewill prove that120572

119899(sdot) convergesweakly to the Brownian

bridge 119861(sdot) with the following covariance structure

1205902(119909 119910) = 119865 (119909 and 119910) minus 119865 (119909) 119865 (119910) for 119909 119910 isin [0 1]

(106)

It turns out that the tightness criterion (8) suffices to reachthe goal Convergence of finite-dimensional distributions of120572

119899(sdot) holds and we shall show itProceeding like Cai and Roussas in [1] we need to show

that for any 119886 119887 isin R

119886120572119899(119909) + 119887120572

119899(119910) 997888rarr 119886119861 (119909) + 119887119861 (119910) in distribution

(107)

Let us introduce the notation

119884119894(119909) = 119870(

119909 minus 119883119894

ℎ119894

) minus 119864119870(119909 minus 119883

119894

ℎ119894

) (108)

and look into the covariance structure of 120572119899(sdot)

Cov (120572119899(119909) 120572

119899(119910))

= Cov( 1

radic119899

119899

sum

119894=1

119884119894(119909)

1

radic119899

119899

sum

119894=1

119884119894(119910))

=1

119899

119899

sum

119894=1

Cov (119884119894(119909) 119884

119894(119910))

=1

119899

119899

sum

119894=1

Cov(119870(119909 minus 119883

119894

ℎ119894

) 119870(119910 minus 119883

119894

ℎ119894

))

(109)

where the second equality follows from assumed indepen-dence of rvrsquos 119883

119894119894ge1

Firstly we observe that each summand converges to119865(119909and

119910) minus 119865(119909)119865(119910) since

Cov(119870(119909 minus 119883

119894

ℎ119894

) 119870(119910 minus 119883

119894

ℎ119894

))

= 119864[119870(119909 minus 119883

119894

ℎ119894

)119870(119910 minus 119883

119894

ℎ119894

)]

minus 119864119870(119909 minus 119883

119894

ℎ119894

)119864119870(119910 minus 119883

119894

ℎ119894

)

(110)

where

119864[119870(119909 minus 119883

119894

ℎ119894

)119870(119910 minus 119883

119894

ℎ119894

)]

= int

119909and119910

minusinfin

119870(119909 minus 119905

ℎ119894

)119870(119910 minus 119905

ℎ119894

)119889119865 (119905)

+ int

119909or119910

119909and119910

119870(119909 minus 119905

ℎ119894

)119870(119910 minus 119905

ℎ119894

)119889119865 (119905)

+ int

infin

119909or119910

119870(119909 minus 119905

ℎ119894

)119870(119910 minus 119905

ℎ119894

)119889119865 (119905)

(111)

and 119865 denotes the common df of the rvrsquos 119883119894119894ge1

In [6] itwas shown that

119864119870(119909 minus 119883

119894

ℎ119894

) 997888rarr 119865 (119909) (112)

and recalling that the kernel function 119870 is a df as well wearrive at the conclusion

Secondly applying Toeplitz lemma we get

Cov (120572119899(119909) 120572

119899(119910)) 997888rarr 119865 (119909 and 119910) minus 119865 (119909) 119865 (119910)

= 1205902(119909 119910) 119899 997888rarr infin

(113)

Thus

Var (119886120572119899(119909) + 119887120572

119899(119910))

= 1198862 Var (120572

119899(119909)) + 2119886119887Cov (120572

119899(119909) 120572

119899(119910))

+ 1198872 Var (120572

119899(119910))

997888rarr 1198862120590

2(119909 119909) + 2119886119887120590

2(119909 119910) + 119887

2120590

2(119910 119910) 119899 997888rarr infin

(114)

We are now on the way to prove that 119886120572119899(119909) + 119887120572

119899(119910) con-

verges in distribution to 119886119861(119909) + 119887119861(119910) sim N(0 1198862120590

2(119909 119909) +

21198861198871205902(119909 119910) + 119887

2120590

2(119910 119910)) which in other words means that

1

radic119899

infin

sum

119894=1

119886119884119894(119909) + 119887119884

119894(119910)

radicVar (119886120572119899(119909) + 119887120572

119899(119910))

997888rarr N (0 1) (115)

In order to obtain (115) we will use Lyapunov condition

lim119899rarrinfin

1

1198632+120575

119899

119899

sum

119894=1

1198641003816100381610038161003816119886119884119894

(119909) + 119887119884119894(119910)

1003816100381610038161003816

2+120575

= 0 (116)

for 120575 = 1 where 1198632

119899= Var(sum119899

119894=1[119886119884

119894(119909) + 119887119884

119894(119910)]) Since

|119870(sdot) minus 119864119870(sdot)| le 2

1198641003816100381610038161003816119886119884119894

(119909) + 119887119884i (119910)1003816100381610038161003816

3

le 8 (1198863+ 119887

3) (117)

Moreover 1198632

119899= 119899Var(119886120572

119899(119909) + 119887120572

119899(119910)) together with (114)

yields

1198633

119899= 119874 (119899

32) (118)

12 ISRN Probability and Statistics

Combining (117) with (118) we obtain

1

1198633

119899

119899

sum

119894=1

1198641003816100381610038161003816119886119884119894

(119909) + 119887119884119894(119910)

1003816100381610038161003816

3

= 119874 (119899minus12

) (119)

which shows that Lyapunov condition holds and completesthe proof of convergence of finite-dimensional distributionsof 120572

119899(119909)We summarize the result of that section in the following

theorem

Theorem 7 Let 120572119899(119909) be the recursive kernel-type empirical

process defined by (105) built on iid rvrsquos 119883119895119895ge1

withmarginal df 119865 If 120572

119899(sdot) satisfies the tightness criterion (8) then

120572119899(sdot) 997888rarr 119861 (sdot) weakly in 119863 [0 1] (120)

where 119861 is the Brownian bridge

Acknowledgment

The author would like to thank the referees for the carefulreading of the paper and helpful remarks

References

[1] Z Cai and G G Roussas ldquoWeak convergence for smoothestimator of a distribution function under negative associationrdquoStochastic Analysis and Applications vol 17 no 2 pp 145ndash1681999

[2] Z Cai andG G Roussas ldquoEfficient Estimation of a DistributionFunction under Qudrant Dependencerdquo Scandinavian Journal ofStatistics vol 25 no 1 pp 211ndash224 1998

[3] ZW Cai and G G Roussas ldquoUniform strong estimation under120572-mixing with ratesrdquo Statistics amp Probability Letters vol 15 no1 pp 47ndash55 1992

[4] G G Roussas ldquoKernel estimates under association stronguniform consistencyrdquo Statistics amp Probability Letters vol 12 no5 pp 393ndash403 1991

[5] Y F Li and S C Yang ldquoUniformly asymptotic normalityof the smooth estimation of the distribution function underassociated samplesrdquo Acta Mathematicae Applicatae Sinica vol28 no 4 pp 639ndash651 2005

[6] Y Li C Wei and S Yang ldquoThe recursive kernel distributionfunction estimator based onnegatively and positively associatedsequencesrdquo Communications in Statistics vol 39 no 20 pp3585ndash3595 2010

[7] H Yu ldquoA Glivenko-Cantelli lemma and weak convergence forempirical processes of associated sequencesrdquo ProbabilityTheoryand Related Fields vol 95 no 3 pp 357ndash370 1993

[8] Q M Shao and H Yu ldquoWeak convergence for weightedempirical processes of dependent sequencesrdquo The Annals ofProbability vol 24 no 4 pp 2098ndash2127 1996

[9] P Billingsley Convergence of Probability Measures Wiley Seriesin Probability and Statistics Probability and Statistics JohnWiley amp Sons New York NY USA 1999

[10] J D Esary F Proschan and D W Walkup ldquoAssociation ofrandom variables with applicationsrdquo Annals of MathematicalStatistics vol 38 pp 1466ndash1474 1967

[11] A Bulinski and A Shashkin Limit Theorems for AssociatedRandom Fields and Related Systems vol 10 of Advanced Serieson Statistical Science and Applied Probability World Scientific2007

[12] S Karlin and Y Rinott ldquoClasses of orderings of measures andrelated correlation inequalities I Multivariate totally positivedistributionsrdquo Journal of Multivariate Analysis vol 10 no 4 pp467ndash498 1980

[13] P Matuła and M Ziemba ldquoGeneralized covariance inequali-tiesrdquo Central European Journal of Mathematics vol 9 no 2 pp281ndash293 2011

[14] P Doukhan and S Louhichi ldquoA new weak dependence condi-tion and applications to moment inequalitiesrdquo Stochastic Proc-esses and their Applications vol 84 no 2 pp 313ndash342 1999

[15] P Matuła ldquoA note on some inequalities for certain classes ofpositively dependent random variablesrdquo Probability and Math-ematical Statistics vol 24 no 1 pp 17ndash26 2004

[16] S Louhichi ldquoWeak convergence for empirical processes of asso-ciated sequencesrdquo Annales de lrsquoInstitut Henri Poincare vol 36no 5 pp 547ndash567 2000

[17] K Joag-Dev and F Proschan ldquoNegative association of randomvariables with applicationsrdquoThe Annals of Statistics vol 11 no1 pp 286ndash295 1983

[18] C M Newman ldquoAsymptotic independence and limit theoremsfor positively and negatively dependent random variablesrdquo inInequalities in Statistics and Probability Y L Tong Ed vol 5pp 127ndash140 Institute ofMathematical StatisticsHayward CalifUSA 1984

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 ISRN Probability and Statistics

Combining (117) with (118) we obtain

1

1198633

119899

119899

sum

119894=1

1198641003816100381610038161003816119886119884119894

(119909) + 119887119884119894(119910)

1003816100381610038161003816

3

= 119874 (119899minus12

) (119)

which shows that Lyapunov condition holds and completesthe proof of convergence of finite-dimensional distributionsof 120572

119899(119909)We summarize the result of that section in the following

theorem

Theorem 7 Let 120572119899(119909) be the recursive kernel-type empirical

process defined by (105) built on iid rvrsquos 119883119895119895ge1

withmarginal df 119865 If 120572

119899(sdot) satisfies the tightness criterion (8) then

120572119899(sdot) 997888rarr 119861 (sdot) weakly in 119863 [0 1] (120)

where 119861 is the Brownian bridge

Acknowledgment

The author would like to thank the referees for the carefulreading of the paper and helpful remarks

References

[1] Z Cai and G G Roussas ldquoWeak convergence for smoothestimator of a distribution function under negative associationrdquoStochastic Analysis and Applications vol 17 no 2 pp 145ndash1681999

[2] Z Cai andG G Roussas ldquoEfficient Estimation of a DistributionFunction under Qudrant Dependencerdquo Scandinavian Journal ofStatistics vol 25 no 1 pp 211ndash224 1998

[3] ZW Cai and G G Roussas ldquoUniform strong estimation under120572-mixing with ratesrdquo Statistics amp Probability Letters vol 15 no1 pp 47ndash55 1992

[4] G G Roussas ldquoKernel estimates under association stronguniform consistencyrdquo Statistics amp Probability Letters vol 12 no5 pp 393ndash403 1991

[5] Y F Li and S C Yang ldquoUniformly asymptotic normalityof the smooth estimation of the distribution function underassociated samplesrdquo Acta Mathematicae Applicatae Sinica vol28 no 4 pp 639ndash651 2005

[6] Y Li C Wei and S Yang ldquoThe recursive kernel distributionfunction estimator based onnegatively and positively associatedsequencesrdquo Communications in Statistics vol 39 no 20 pp3585ndash3595 2010

[7] H Yu ldquoA Glivenko-Cantelli lemma and weak convergence forempirical processes of associated sequencesrdquo ProbabilityTheoryand Related Fields vol 95 no 3 pp 357ndash370 1993

[8] Q M Shao and H Yu ldquoWeak convergence for weightedempirical processes of dependent sequencesrdquo The Annals ofProbability vol 24 no 4 pp 2098ndash2127 1996

[9] P Billingsley Convergence of Probability Measures Wiley Seriesin Probability and Statistics Probability and Statistics JohnWiley amp Sons New York NY USA 1999

[10] J D Esary F Proschan and D W Walkup ldquoAssociation ofrandom variables with applicationsrdquo Annals of MathematicalStatistics vol 38 pp 1466ndash1474 1967

[11] A Bulinski and A Shashkin Limit Theorems for AssociatedRandom Fields and Related Systems vol 10 of Advanced Serieson Statistical Science and Applied Probability World Scientific2007

[12] S Karlin and Y Rinott ldquoClasses of orderings of measures andrelated correlation inequalities I Multivariate totally positivedistributionsrdquo Journal of Multivariate Analysis vol 10 no 4 pp467ndash498 1980

[13] P Matuła and M Ziemba ldquoGeneralized covariance inequali-tiesrdquo Central European Journal of Mathematics vol 9 no 2 pp281ndash293 2011

[14] P Doukhan and S Louhichi ldquoA new weak dependence condi-tion and applications to moment inequalitiesrdquo Stochastic Proc-esses and their Applications vol 84 no 2 pp 313ndash342 1999

[15] P Matuła ldquoA note on some inequalities for certain classes ofpositively dependent random variablesrdquo Probability and Math-ematical Statistics vol 24 no 1 pp 17ndash26 2004

[16] S Louhichi ldquoWeak convergence for empirical processes of asso-ciated sequencesrdquo Annales de lrsquoInstitut Henri Poincare vol 36no 5 pp 547ndash567 2000

[17] K Joag-Dev and F Proschan ldquoNegative association of randomvariables with applicationsrdquoThe Annals of Statistics vol 11 no1 pp 286ndash295 1983

[18] C M Newman ldquoAsymptotic independence and limit theoremsfor positively and negatively dependent random variablesrdquo inInequalities in Statistics and Probability Y L Tong Ed vol 5pp 127ndash140 Institute ofMathematical StatisticsHayward CalifUSA 1984

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of