Research Article A General Result on the Mean Integrated ...downloads.hindawi.com/journals/jps/2014/403764.pdf · Research Article A General Result on the Mean Integrated Squared

Research ArticleA General Result on the Mean Integrated SquaredError of the Hard Thresholding Wavelet Estimator under120572-Mixing Dependence

Christophe Chesneau

Laboratoire de Matheematiques Nicolas Oresme Universite de Caen BP 5186 14032 Caen Cedex France

Correspondence should be addressed to Christophe Chesneau christophechesneaugmailcom

Received 10 April 2013 Accepted 22 October 2013 Published 19 January 2014

Academic Editor Zhidong Bai

Copyright copy 2014 Christophe Chesneau This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

We consider the estimation of an unknown function119891 for weakly dependent data (120572-mixing) in a general setting Our contributionis theoretical we prove that a hard thresholding wavelet estimator attains a sharp rate of convergence under the mean integratedsquared error (MISE) over Besov balls without imposing too restrictive assumptions on the model Applications are given for twotypes of inverse problems the deconvolution density estimation and the density estimation in a GARCH-typemodel both improveexisting results in this dependent context Another application concerns the regression model with random design

1 Introduction

A general nonparametric problem is adopted we aim toestimate an unknown function 119891 via 119899 random variables119881

1 119881

119899from a strictly stationary stochastic process (119881

119905)119905isinZ

We suppose that (119881119905)119905isinZ has a weak dependence structure

the 120572-mixing case is considered This kind of dependencenaturally appears in numerous models as Markov chainsGARCH-type models and discretely observed diffusions(see eg [1ndash3]) The problems where 119891 is the density of 119881

1

or a regression function have received a lot of attention Apartial list of related works includes Robinson [4] Roussas[5 6] Truong and Stone [7] Tran [8] Masry [9 10] Masryand Fan [11] Bosq [12] and Liebscher [13]

For an efficient estimation of 119891 many methods canbe considered The most popular of them are based onkernels splines and wavelets In this note we deal withwavelet methods that have been introduced in iidsettingby Donoho and Johnstone [14 15] and Donoho et al [1617] These methods enjoy remarkable local adaptivity againstdiscontinuities and spatially varying degree of oscillationsComplete reviews and discussions on wavelets in statisticscan be found in for example Antoniadis [18] and Hardleet al [19] In the context of 120572-mixing dependence various

wavelet methods have been elaborated for a wide variety ofnonparametric problems Recent developments can be foundin for example Leblanc [20] Tribouley and Viennet [21]Masry [22] Patil and Truong [23] Doosti et al [24] Doostiand Niroumand [25] Doosti et al [26] Cai and Liang [27]Niu and Liang [28] Benatia and Yahia [29] Chesneau [30ndash32] Chaubey and Shirazi [33] and Abbaszadeh and Emadi[34]

In the general dependent setting described above weprovide a theoretical contribution to the performance ofa wavelet estimator based on a hard thresholding Thisnonlinear wavelet procedure has the features to be fullyadaptive and efficient over a large class of functions 119891 (seeeg [14ndash17 35]) Following the spirit of Kerkyacharian andPicard [36] we determine necessary assumptions on (119881

119905)119905isinZ

and the wavelet basis to ensure that the considered estimatorattains a fast rate of convergence under the MISE over Besovballs The obtained rate of convergence often corresponds tothe near optimal one in the minimax sense for the standardiid case The originality of our result is to be general andsharp it can be applied for nonparametricmodels of differentnatures and improves some existing results This fact isillustrated by the consideration of the density deconvolutionestimation problem and the density estimation problem in a

Hindawi Publishing CorporationJournal of Probability and StatisticsVolume 2014 Article ID 403764 12 pageshttpdxdoiorg1011552014403764

2 Journal of Probability and Statistics

GARCH-type model improving ([30] Proposition 51) and([31] Theorem 2) respectively A last part is devoted to theregression model with random design The obtained resultcompletes the one of Patil and Truong [23]

The organization of this note is as follows In the nextsection we describe the considered wavelet setting Thehard thresholding estimator and its rate of convergenceunder the MISE over Besov balls are presented in Section 3Applications of our general result are given in Section 4 Theproofs are carried out in Section 5

2 Wavelets and Besov Balls

In this section we introduce some notations corresponding towavelets and Besov balls

21 Wavelet Basis We consider the wavelet basis on [0 1]constructs from the Daubechies wavelets db2N with 119873 ge 1(see eg [37]) A brief description of this basis is given belowLet 120601 and 120595 be the initial wavelet functions of the familydb2N These functions have the particularity to be compactlysupported and to belong to the class C119886 for 119873 gt 5119886 For any119895 ge 0 we set Λ

119895= 0 2119895 minus 1 and for 119896 isin Λ

119895

120601119895119896

(119909) = 21198952120601 (2119895119909 minus 119896) 120595119895119896

(119909) = 21198952120595 (2119895119909 minus 119896)

(1)

With appropriated treatments at the boundaries thereexists an integer 120591 such that for any integer ℓ ge 120591 B =120601

ℓ119896 119896 isin Λ

ℓ 120595

119895119896 119895 isin N minus 0 ℓ minus 1 119896 isin Λ

119895 is an

orthonormal basis of L2([0 1]) where

L2([0 1])

= 119891 [0 1] 997888rarr R1003817100381710038171003817119891

10038171003817100381710038172 = (int1

0

1003816100381610038161003816119891 (119909)10038161003816100381610038162

119889119909)

12

lt infin

(2)

For any integer ℓ ge 120591 and 119891 isin L2([0 1]) we have thefollowing wavelet expansion

119891 (119909) = sum119896isinΛ ℓ

119888ℓ119896

120601ℓ119896

(119909) +infin

sum119895=ℓ

sum119896isinΛ 119895

119889119895119896

120595119895119896

(119909)

119909 isin [0 1]

(3)

where 119888119895119896

and 119889119895119896

denote the wavelet coefficients of119891 definedby

119888119895119896

= int1

0

119891 (119909) 120601119895119896

(119909) 119889119909 119889119895119896

= int1

0

119891 (119909) 120595119895119896

(119909) 119889119909

(4)

Technical details can be found in for example Cohen et al[38] and Mallat [39]

In the main result of this paper we will investigate theMISE rate of the proposed estimator by assuming that theunknown function of interest 119891 belongs to a wide class offunctions the Besov class Its definition in terms of waveletcoefficients is presented in the following

22 Besov Balls We say that 119891 isin 119861119904

119901119903(119872)with 119904 gt 0 119901 119903 ge 1

and 119872 gt 0 if and only if there exists a constant 119862 gt 0 suchthat the wavelet coefficients of 119891 given by (4) satisfy

2120591(12minus1119901)( sum119896isinΛ 120591

10038161003816100381610038161198881205911198961003816100381610038161003816119901

)

1119901

+ (infin

sum119895=120591

(2119895(119904+12minus1119901)( sum119896isinΛ 119895

10038161003816100381610038161003816119889119895119896

10038161003816100381610038161003816119901

)

1119901

)

119903

)

1119903

le 119862

(5)

with the usualmodifications if119901 = infin or 119903 = infin Note that forparticular choices of 119904 119901 and 119903119861119904

119901119903(119872) contains the classical

Holder and Sobolev balls (see eg [40] and [19])

Remark 1 We have chosen a wavelet basis on [0 1] to fix thenotations wavelet basis on another interval can be consideredin the rest of the study without affecting the results

3 Statistical Framework Estimator and Result

31 Statistical Framework As mentioned in Section 1 anonparametric estimation setting as general as possible isadopted we aim to estimate an unknown function 119891 isin

L2([0 1]) via 119899 random variables (or vectors) 1198811 119881

119899from

a strictly stationary stochastic process (119881119905)119905isinZ defined on a

probability space (ΩAP) We suppose that (119881119905)119905isinZ has a

120572-mixing dependence structure with exponential decay ratethat is there exist two constants 120574 gt 0 and 120579 gt 0 such that

sup119898ge1

(119890120579119898120572119898) le 120574 (6)

where 120572119898

= sup(119860119861)isinF119881

minusinfin0timesF119881119898infin

|P(119860 cap 119861) minus P(119860)P(119861)|F119881

minusinfin0is the 120590-algebra generated by the random variables (or

vectors) 119881minus1

1198810and F119881

119898infinis the 120590-algebra generated by

the random variables (or vectors) 119881119898 119881

119898+1

The 120572-mixing dependence is reasonably weak it is sat-isfied by a wide variety of models including Markov chainsGARCH-type models and discretely observed diffusions(see for instance [1ndash3 41])

The considered estimator for 119891 is presented below

32 Estimator We define the hard thresholding waveletestimator 119891 by

119891 (119909) = sum119896isinΛ 1198950

1198881198950 119896

1206011198950 119896

(119909) +

1198951

sum119895=1198950

sum119896isinΛ 119895

1198891198951198961|119889119895119896|ge120581120582119895

120595119895119896

(119909)

(7)

where

119888119895119896

=1

119899

119899

sum119894=1

119902 (120601119895119896

119881119894) 119889

119895119896=

1

119899

119899

sum119894=1

119902 (120595119895119896

119881119894) (8)

Journal of Probability and Statistics 3

1 is the indicator function 120581 gt 0 is a large enough constant1198950is the integer satisfying

21198950 = [120591 ln 119899] (9)

where [119886] denotes the integer part of 119886 and 1198951is the integer

satisfying

21198951 = [(119899

(ln 119899)3)

1(2120588+1)

] (10)

120582119895= 2120588119895radic ln 119899

119899 (11)

Here it is supposed that there exists a function 119902 L2([0 1])times119881

1(Ω) rarr C such that

(H1) for 120574 isin 120601 120595 any integer 119895 ge 1198950and 119896 isin Λ

119895

E (119902 (120574119895119896

1198811)) = int

1

0

119891 (119909) 120574119895119896

(119909) 119889119909 (12)

where E denotes the expectation(H2) there exist two constants 119862 gt 0 and 120588 ge 0

satisfying for 120574 isin 120601 120595 for any integer 119895 ge 1198950

and 119896 isin Λ119895 (i) sup

119909isin1198811(Ω)|119902(120574

119895119896 119909)| le 119862212058811989521198952 (ii)

E(|119902(120574119895119896

1198811)|2) le 11986222120588119895 (iii) for any 119898 isin 1 119899 minus

1 ge 110038161003816100381610038161003816C119900V (119902 (120574

119895119896 119881

119898+1) 119902 (120574

119895119896 119881

1))

10038161003816100381610038161003816 le 119862221205881198952minus119895 (13)

whereC119900V denotes the covariance that isC119900V(119883 119884) =

E(119883119884)minusE(119883)E(119884) 119884 denotes the complex conjugateof 119884

For well-known nonparametric models in the iid set-ting hard thresholding wavelet estimators and importantresults can be found in for example Donoho and Johnstone[14 15] Donoho et al [16 17] Delyon and Juditsky [35]Kerkyacharian and Picard [36] and Fan and Koo [42] In the120572-mixing context 119891 defined by (7) is a general and improvedversion of the estimator considered in Chesneau [30 31] Themain differences are the presence of the tuning parameter 120588and the global definition of the function 119902 offering numerouspossibilities of applications Three of them are explored inSection 4

Comments on the Assumptions The assumption (H1) ensuresthat (8) are unbiased estimators for 119888

119895119896and 119889

119895119896given by

(4) whereas (H2) is related to their good performance SeeProposition 10These assumptions are not too restrictive Forinstance if we consider the standard density estimation prob-lem where (119881

119905)119905isinZ are iid random variables with bounded

density 119891 the function 119902(120574 119909) = 120574(119909) satisfies (H1) and (H2)with 120588 = 0 (note that thanks to the independence of (119881

119905)119905isinZ

the covariance term in (H2)-(iii) is zero)The technical detailsare given in Donoho et al [17]

Lemma 2 describes a simple situation in which assump-tion (H2)-(iii) is satisfied

Lemma 2 We make the following assumptions

(F1) Let119906 be the density of 1198811and let119906

(1198811 119881119898+1)be the density

of (1198811 119881

119898+1) for any 119898 isin Z We suppose that there

exists a constant 119862 gt 0 such that

sup119898isin1119899minus1Z

sup(119909119910)isin1198811(Ω)times119881119898+1(Ω)

10038161003816100381610038161003816119906(1198811 119881119898+1)(119909 119910)

minus119906 (119909) 119906 (119910)1003816100381610038161003816 le 119862

(14)

(F2) There exist two constants 119862 gt 0 and 120588 ge 0 satisfyingfor 120574 isin 120601 120595 for any integer 119895 ge 119895

0and 119896 isin Λ

119895

int1198811(Ω)

10038161003816100381610038161003816119902 (120574119895119896

119909)10038161003816100381610038161003816 119889119909 le 11986221205881198952minus1198952 (15)

Then under (F1) and (F2) (H2)-(iii) is satisfied

33 Result Theorem 3 determines the rate of convergenceattained by 119891 under the MISE over Besov balls

Theorem 3 We consider the general statistical settingdescribed in Section 31 Let 119891 be (7) under (H1) and (H2)Suppose that 119891 isin 119861119904

119901119903(119872) with 119903 ge 1 119901 ge 2 and 119904 isin (0119873)

or 119901 isin [1 2) and 119904 isin ((2120588 + 1)119901119873) Then there exists aconstant 119862 gt 0 such that

E (10038171003817100381710038171003817119891 minus 119891

100381710038171003817100381710038172

2) le 119862(

ln 119899

119899)

2119904(2119904+2120588+1)

(16)

The rate of convergence ldquo((ln 119899)119899)2119904(2119904+2120588+1)rdquo is oftenthe near optimal one in the minimax sense for numerousstatistical problems in a iid setting (see eg [19 43])Moreover note that Theorem 3 is flexible the assumptionson (119881

119905)119905isinZ related to the definition of 119902 in (H1) and (H2)

are mild In the next section this flexibility is illustrated forthree sophisticated nonparametric estimation problems thedensity deconvolution estimation problem the density esti-mation problem in a GARCH-typemodel and the regressionfunction estimation in the regression model with randomdesign

4 Applications

41 Density Deconvolution Let (119881119905)119905isinZ be a strictly stationary

stochastic process such that

119881119905= 119883

119905+ 120598

119905 119905 isin Z (17)

where (119883119905)119905isinZ is a strictly stationary stochastic process with

unknown density119891 and (120598119905)119905isinZ is a strictly stationary stochas-

tic process with known density 119892 It is supposed that 120598119905and

119883119905are independent for any 119905 isin Z and (119881

119905)119905isinZ is a 120572-mixing

process with exponential decay rate (see Section 31 for aprecise definition) Our aim is to estimate 119891 via 119881

1 119881

119899

from (119881119905)119905isinZ Some related works are Masry [44] Kulik [45]

Comte et al [46] and Van Zanten and Zareba [47]We formulate the following assumptions


(G1) The support of 119891 is [0 1](G2) There exists a constant 119862 gt 0 such that

sup119909isinR

119891 (119909) le 119862 lt infin (18)

(G3) Let 119906 be the density of1198811We suppose that there exists

a constant 119862 gt 0 such that

sup119909isinR

119906 (119909) le 119862 (19)

(G4) For any 119898 isin Z let 119906(1198811 119881119898+1)

be the density of(119881

1 119881

119898+1) We suppose that there exists a constant

119862 gt 0 such that

sup119898isinZ

sup(119909119910)isinR2

119906(1198811 119881119898+1)

(119909 119910) le 119862 (20)

(G5) For any integrable function 120574 we define its Fouriertransform by

F (120574) (119909) = intinfin

minusinfin

120574 (119910) 119890minus119894119909119910119889119910 119909 isin R (21)

We suppose that there exist three known constants119862 gt 0119888 gt 0 and 120575 gt 1 such that for any 119909 isin R

(i) the Fourier transform of 119892 satisfies

1003816100381610038161003816F (119892) (119909)1003816100381610038161003816 ge

119888

(1 + 1199092)1205752

(22)

(ii) for any ℓ isin 0 1 2 the ℓth derivative of the Fouriertransform of 119892 satisfies

100381610038161003816100381610038161003816(F (119892) (119909))

(ℓ)100381610038161003816100381610038161003816le

119862

(1 + |119909|)120575+ℓ (23)

We are now in the position to present the result

Theorem 4 We consider the model (17) Suppose that (G1)ndash(G5) are satisfied Let 119891 be defined as in (7) with

119902 (120574 119909) =1

2120587int

infin

minusinfin

F (120574) (119910)

F (119892) (119910)119890minus119894119910119909119889119910 (24)

whereF(120574)(119910) denotes the complex conjugate ofF(120574)(119910) and120588 = 120575 (appearing in(G5))

Suppose that 119891 isin 119861119904

119901119903(119872) with 119903 ge 1 119901 ge 2 and 119904 isin

(0119873) or 119901 isin [1 2) and 119904 isin ((2120575 + 1)119901119873) Then thereexists a constant 119862 gt 0 such that

E (10038171003817100381710038171003817119891 minus 119891

100381710038171003817100381710038172

2) le 119862(

ln 119899

119899)

2119904(2119904+2120575+1)

(25)

Theorem 4 improves ([30] Proposition 51) in terms ofrate of convergence we gain a logarithmic term

Moreover it is established that in the iid settingldquo((ln 119899)119899)2119904(2119904+2120575+1)rdquo is

(i) exactly the rate of convergence attained by the hardthresholding wavelet estimator

(ii) the near optimal rate of convergence in the minimaxsense

The details can be found in Fan and Koo [42] ThusTheorem 4 can be viewed as an extension of this existingresult to the weak dependent case

42 GARCH-TypeModel We consider the strictly stationarystochastic process (119881

119905)119905isinZ where for any 119905 isin Z

119881119905= 1205902

119905119885

119905 (26)

(1205902

119905)119905isinZ is a strictly stationary stochastic process with

unknown density 119891 and (119885119905)119905isinZ is a strictly stationary

stochastic process with known density 119892 It is supposed that1205902

119905and 119885


119905)119905isinZ is a 120572-

mixing process with exponential decay rate (see Section 31for a precise definition) Our aim is to estimate 119891 via119881

1 119881

119899from (119881

119905)119905isinZ Some related works are Comte et al

[46] and Chesneau [31]We formulate the following assumptions(J1) There exists a positive integer 120575 such that

119892 (119909) =1

(120575 minus 1)(minus ln119909)

120575minus1 119909 isin [0 1] (27)

Let us remark that 119892 is the density of prod120575

119894=1119880

119894 where

1198801 119880

120575are 120575 iid random variables having the

common distributionU([0 1])(J2) The support of 119891 is [0 1] and 119891 isin L2([0 1])(J3) Let 119906 be the density of119881

1We suppose that there exists

a constant 119862 gt 0 such thatsup119909isinR

119906 (119909) le 119862 (28)

(J4) For any 119898 isin Z let 119906(1198811 119881119898+1)


1 119881


119862 gt 0 such thatsup119898isinZ

sup(119909119910)isinR2

119906(1198811 119881119898+1)

(119909 119910) le 119862 (29)


Theorem 5 We consider model (26) Suppose that (J1)ndash(J4)are satisfied Let 119891 be defined as in (7) with

119902 (120574 119909) = 119879120575(120574) (119909) (30)

where for any positive integer ℓ 119879(120574)(119909) = (119909120574(119909))1015840 and119879

ℓ(120574)(119909) = 119879(119879

ℓminus1(120574))(119909) and 120588 = 120575 (appearing in (J1))




E (10038171003817100381710038171003817119891 minus 119891

100381710038171003817100381710038172

2) le 119862(

ln 119899

119899)

2119904(2119904+2120575+1)

(31)

Theorem 5 significantly improves ([31] Theorem 2) interms of rate of convergence we gain an exponent 12


43 Nonparametric Regression Model We consider thestrictly stationary stochastic process (119881

119905)119905isinZ where for any

119905 isin Z 119881119905= (119884

119905 119883

119905)

119884119905= 119891 (119883

119905) + 120585

119905 (32)

(119883119905)119905isinZ is a strictly stationary stochastic process with

unknown density 119892 (120585119905)119905isinZ is a strictly stationary centered

stochastic process and119891 is the unknown regression functionIt is supposed that 119883

119905and 120585

119905are independent for any 119905 isin Z

and (119881119905)119905isinZ is a 120572-mixing process with exponential decay

rate (see Section 31 for a precise definition) Our aim is toestimate 119891 via 119881

1 119881

119899from (119881

119905)119905isinZ Applications of this

problem can be found in Hardle [48] Wavelet methods canbe found in Patil and Truong [23] Doosti et al [24] Doostiet al [26] and Doosti and Niroumand [25]

We formulate the following assumptions

(K1) The support of 119891 and 119892 is [0 1] and 119891 and 119892 isin

L2([0 1])(K2) 120585

1(Ω) is bounded

(K3) There exists a constant 119862 gt 0 such that

sup119909isin[01]

1003816100381610038161003816119891 (119909)1003816100381610038161003816 le 119862 (33)

(K4) There exist two constants 119888lowastgt 0 and 119862 gt 0 such that

119888lowastle inf

119909isin[01]

119892 (119909) sup119909isin[01]

119892 (119909) le 119862 (34)

(K5) Let 119906 be the density of1198811We suppose that there exists


sup119909isinRtimes[01]

119906 (119909) le 119862 (35)

(K6) For any 119898 isin Z let 119906(1198811 119881119898+1)


1 119881



sup119898isinZ

sup(119909119910)isin(Rtimes[01])times(Rtimes[01])

119906(1198811 119881119898+1)

(119909 119910) le 119862 (36)


Theorem 6 We consider the model (32) Suppose that (K1)ndash(K6) are satisfied Let 119891 be the truncated ratio estimatorConsider

119891 (119909) =V (119909)119892 (119909)

1|119892(119909)|ge119888lowast2

(37)

where

(i) V is defined as in (7) with

119902 (120574 (119909 119909lowast)) = 119909120574 (119909

lowast) (38)

and 120588 = 0

(ii) 119892 is defined as in (7) with 119883119905instead of 119881

119905

119902 (120574 119909) = 120574 (119909) (39)

and 120588 = 0(iii) 119888

lowastis the constant defined in (K4)


119901119903(119872) and 119892 isin 119861119904

119901119903(119872) with 119903 ge 1

119901 ge 2 and 119904 isin (0119873) or 119901 isin [1 2) and 119904 isin (1119901119873) Thenthere exists a constant 119862 gt 0 such that

E (10038171003817100381710038171003817119891 minus 119891

100381710038171003817100381710038172

2) le 119862(

ln 119899

119899)

2119904(2119904+1)

(40)

The estimator (37) is derived by combining the procedureof Patil and Truong [23] with the truncated approach ofVasiliev [49]

Theorem 6 completes Patil and Truong [23] in terms ofrates of convergence under the MISE over Besov balls

Remark 7 The assumption(K2) can be relaxed with anotherstrategy to the one developed in Theorem 6 Some technicalelements are given in Chesneau [32]

Conclusion Considering the weak dependent case on theobservations we prove a general result on the rate of conver-gence attains by a hard wavelet thresholding estimator underthe MISE over Besov balls This result is flexible it can beapplied for a wide class of statistical models Moreover theobtained rate of convergence is sharp it can correspond to thenear optimal one in the minimax sense for the standard iidcase Some recent results on sophisticated statistical problemsare improved Thanks to its flexibility the perspectives ofapplications of our theoretical result in other contexts arenumerous

5 Proofs

In this section 119862 denotes any constant that does not dependon 119895 119896 and 119899 Its value may change from one term to anotherand may depend on 120601 or 120595

51 Key Lemmas Let us present two lemmas which will beused in the proofs

Lemma 8 shows a sharp covariance inequality under the120572-mixing condition

Lemma 8 (see [50]) Let (119882119905)119905isinZ be a strictly stationary 120572-

mixing process withmixing coefficient 120572119898119898 ge 0 and let ℎ and

119896 be two measurable functions Let 119901 gt 0 and 119902 gt 0 satisfying1119901 + 1119902 lt 1 such that E(|ℎ(119882

1)|119901) and E(|119896(119882

1)|119902) exist

Then there exists a constant 119862 gt 0 such that1003816100381610038161003816C119900V (ℎ (119882

1) 119896 (119882

119898+1))1003816100381610038161003816

le 1198621205721minus1119901minus1119902

119898(E (

1003816100381610038161003816ℎ (1198821)1003816100381610038161003816119901

))1119901

(E(1003816100381610038161003816119896 (119882

1)1003816100381610038161003816119902

))1119902

(41)

Lemma 9 below presents a concentration inequality for120572-mixing processes


Lemma9 (see [13]) Let (119882119905)119905isinZ be a strictly stationary process

with the 119898th strongly mixing coefficient 120572119898 119898 ge 0 let 119899 be a

positive integer let ℎ R rarr C be a measurable function andfor any 119905 isin Z 119880

119905= ℎ(119882

119905) We assume that E(119880

1) = 0 and

there exists a constant 119872 gt 0 satisfying |1198801| le 119872 Then for

any 119898 isin 1 [1198992] and 120582 gt 0 we have

P(1

119899

100381610038161003816100381610038161003816100381610038161003816

119899

sum119894=1

119880119894

100381610038161003816100381610038161003816100381610038161003816ge 120582)

le 4 exp(minus1205822119899

16 (119863119898119898 + 1205821198721198983)

) + 32119872

120582119899120572

119898

(42)

where

119863119898

= max119897isin12119898

V (119897

sum119894=1

119880119894) (43)

52 Intermediary Results

Proof of Lemma 2 Using a standard expression of the covari-ance and (F1) as well as(F2) we obtain

10038161003816100381610038161003816C119900V (119902 (120574119895119896

119881119898+1

) 119902 (120574119895119896

1198811))

10038161003816100381610038161003816

=100381610038161003816100381610038161003816100381610038161003816int1198811(Ω)

int1198811(Ω)

119902 (120574119895119896

119909) 119902 (120574119895119896

119910)

times (119906(1198811 119881119898+1)

(119909 119910) minus 119906 (119909) 119906 (119910)) 119889119909119889119910100381610038161003816100381610038161003816100381610038161003816

le int1198811(Ω)

int1198811(Ω)

10038161003816100381610038161003816119902 (120574119895119896

119909)1003816100381610038161003816100381610038161003816100381610038161003816119902 (120574

119895119896 119910)

10038161003816100381610038161003816

times10038161003816100381610038161003816119906(1198811 119881119898+1)

(119909 119910) minus 119906 (119909) 119906 (119910)10038161003816100381610038161003816 119889119909 119889119910

le 119862(int1198811(Ω)

10038161003816100381610038161003816119902 (120574119895119896

119909)10038161003816100381610038161003816 119889119909)

2

le 119862221205881198952minus119895

(44)

This ends the proof of Lemma 2

Proposition 10 proves probability and moments inequal-ities satisfied by the estimators (8)

Proposition 10 Let 119895119896

and 120573119895119896

be defined as in (8) under(H1) and (H2) let 119895

0be (9) and let 119895

1be (10)

(a) There exists a constant 119862 gt 0 such that for any 119895 isin119895

0 119895

1 and 119896 isin Λ

119895

E (10038161003816100381610038161003816119888119895119896 minus 119888

119895119896

100381610038161003816100381610038162

) le 11986222120588119895 1

119899 (45)

E (10038161003816100381610038161003816119889119895119896

minus 119889119895119896

100381610038161003816100381610038162

) le 11986222120588119895 1

119899 (46)

(b) There exists a constant 119862 gt 0 such that for any 119895 isin119895

0 119895


119895

E (10038161003816100381610038161003816119889119895119896

minus 119889119895119896

100381610038161003816100381610038164

) le 11986224120588119895 (47)

(c) Let 120582119895be defined as in (11) There exists a constant 119862 gt

0 such that for any 120581 large enough 119895 isin 1198950 119895

1 and

119896 isin Λ119895 we have

P(10038161003816100381610038161003816119889119895119896

minus 119889119895119896

10038161003816100381610038161003816 ge120581120582

119895

2) le 119862

1

1198994 (48)

Proof of Proposition 10 (a) Using (H1) and the stationarity of(119881

119905)119905isinZ we obtain

E (10038161003816100381610038161003816119888119895119896 minus 119888

119895119896

100381610038161003816100381610038162

) = V (119888119895119896

)

=1

1198992

119899

sum119894=1

V (119902 (120601119895119896

119881119894)) +

2

1198992

times119899

sumV=2

Vminus1

sumℓ=1

Re (C119900V (119902 (120601

119895119896 119881V) 119902 (120601

119895119896 119881

ℓ)))

=1

1198992

119899

sum119894=1

V (119902 (120601119895k 119881119894

)) +2

1198992

times119899minus1

sum119898=1

(119899 minus 119898)Re (C119900V (119902 (120601

119895119896 119881

119898+1) 119902 (120601

119895119896 119881

1)))

le1

119899(E (

10038161003816100381610038161003816119902 (120601119895119896

1198811)100381610038161003816100381610038162

)

+2119899minus1

sum119898=1

10038161003816100381610038161003816C119900V (119902 (120601119895119896

119881119898+1

) 119902 (120601119895119896

1198811))

10038161003816100381610038161003816)

(49)

By (H2)-(ii) we get

E (10038161003816100381610038161003816119902(120601119895119896

1198811)100381610038161003816100381610038162

) le 11986222120588119895 (50)

For the covariance term note that

119899minus1

sum119898=1

10038161003816100381610038161003816C119900V (119902 (120601119895119896

119881119898+1

) 119902 (120601119895119896

1198811))

10038161003816100381610038161003816 = 119860 + 119861 (51)

where

119860 =[ln 119899120579]minus1

sum119898=1

10038161003816100381610038161003816C119900V (119902 (120601119895119896

119881119898+1

) 119902 (120601119895119896

1198811))

10038161003816100381610038161003816

119861 =119899minus1

sum119898=[(ln 119899)120579]

10038161003816100381610038161003816C119900V (119902 (120601119895119896

119881119898+1

) 119902 (120601119895119896

1198811))

10038161003816100381610038161003816

(52)

It follows from (H2)-(iii) and 2minus119895 le 2minus1198950 lt 2(ln 119899)minus1 that

119860 le 119862221205881198952minus119895 [ln 119899

120579] le 11986222120588119895 (53)


The Davydov inequality described in Lemma 8 with 119901 = 119902 =

4 (H2)-(i)-(ii) and 2119895 le 21198951 le 119899 give

119861 le 119862radicE (10038161003816100381610038161003816119902(120601119895119896

1198811)100381610038161003816100381610038164

)119899minus1

sum119898=[(ln 119899)120579]

radic120572119898

le 119862212058811989521198952radicE (10038161003816100381610038161003816119902(120601119895119896

1198811)100381610038161003816100381610038162

)infin

sum119898=[(ln 119899)120579]

119890minus1205791198982

= 11986222120588119895radic119899119890minus(ln 119899)2 le 11986222120588119895

(54)

Thus119899minus1

sum119898=1

10038161003816100381610038161003816C119900V (119902 (120601119895119896

119881119898+1

) 119902 (120601119895119896

1198811))

10038161003816100381610038161003816 le 11986222120588119895 (55)

Putting (49) (50) and (55) together the first point in (a)is proved The proof of the second point is identical with 120595instead of 120601

(b) Thanks to (H2)-(i) we have |119889119895119896

| le

sup119909isin1198811(Ω)

|119902(120595119895119896

119909)| le 119862212058811989521198952 It follows from thetriangular inequality and |119889

119895119896| le 119891

2le 119862 that

10038161003816100381610038161003816119889119895119896minus 119889

119895119896

10038161003816100381610038161003816 le10038161003816100381610038161003816119889119895119896

10038161003816100381610038161003816 +10038161003816100381610038161003816119889119895119896

10038161003816100381610038161003816 le 119862212058811989521198952 (56)

This inequality and the second result of (a) yield

E (10038161003816100381610038161003816119889119895119896

minus 119889119895119896

100381610038161003816100381610038164

) le 119862221205881198952119895E (

10038161003816100381610038161003816119889119895119896minus 119889

119895119896

100381610038161003816100381610038162

) le 119862241205881198952119895 1

119899

(57)

Using 2119895 le 21198951 le 119899 the proof of (b) is completed(c) We will use the Liebscher inequality described in

Lemma 9 Let us set

119880119894= 119902 (120595

119895119896 119881

119894) minus E (119902 (120595

119895119896 119881

1)) (58)

We have E(1198801) = 0 and by(H2)-(i) and 2119895 le 21198951 le

119899(ln 119899)3

1003816100381610038161003816119880119894

1003816100381610038161003816 le 2 sup119909isin1198811(Ω)

10038161003816100381610038161003816119902 (120595119895119896

119909)10038161003816100381610038161003816 le 119862212058811989521198952 le 1198622120588119895

radic119899

(ln 119899)3

(59)

(so 119872 = 1198622120588119895radic119899(ln 119899)3)Proceeding as for the proofs of the bounds in (a) for any

integer 119897 le 119862 ln 119899 since 2minus119895 le 2minus1198950 le 2(ln 119899)minus1 we show that

V (119897

sum119894=1

119880119894)

= V (119897

sum119894=1

119902 (120595119895119896

119881119894)) le 11986222120588119895 (119897 + 11989722minus119895) le 11986222120588119895119897

(60)

Therefore

119863119898

= max119897isin12119898

V (119897

sum119894=1

119880119894) le 11986222120588119895119898 (61)

Owing to Lemma 9 applied with 1198801 119880

119899 120582 = 120581120582

1198952

119898 = [radic120581 ln 119899] 119872 = 1198622120588119895radic119899(ln 119899)3 and the bound (61) weobtain

P(10038161003816100381610038161003816119889119895119896

minus 119889119895119896

10038161003816100381610038161003816 ge120581120582

119895

2)

le 119862(exp(minus11986212058121205822

119895119899

119863119898119898 + 120581120582

119895119898119872

) +119872

120582119895

119899119890minus120579119898)

le 119862( exp (minus119862((120581222120588119895 ln 119899)

times (22120588119895 + 1205812120588119895radic (ln 119899)

119899

times [radic120581 ln 119899] 2120588119895

radic119899

(ln 119899)3)

minus1

))

+ radic119899(ln 119899)3

(ln 119899) 119899119899119890minus120579[radic120581 ln 119899])

le 119862(119899minus1198621205812(1+120581

32) + 1198992minus120579radic120581)

(62)

Taking 120581 large enough the last term is bounded by1198621198994Thiscompletes the proof of(c)

This completes the proof of Proposition 10

Proof of Theorem 3 Theorem 3 can be proved by combiningarguments of ([36] Theorem 51) and ([51] Theorem 42) Itis close to ([30] Proof of Theorem 2) by taking 120579 rarr infin Theinterested reader can find the details below

We consider the following wavelet decomposition for 119891

119891 (119909) = sum119896isinΛ 1198950

1198881198950 119896

1206011198950 119896

(119909) +infin

sum119895=1198950

sum119896isinΛ 119895

119889119895119896

120595119895119896

(119909) (63)

where 1198881198950 119896

= int1

0119891(119909)120601

1198950 119896(119909)119889119909 and 119889

119895119896= int

1

0119891(119909)120595

119895119896(119909)119889119909

Using the orthonormality of the wavelet basis B theMISE of 119891 can be expressed as

E (10038171003817100381710038171003817119891 minus 119891

100381710038171003817100381710038172

2) = 119875 + 119876 + 119877 (64)

where

119875 = sum119896isinΛ 1198950

E (100381610038161003816100381610038161198881198950 119896 minus 119888

1198950 119896

100381610038161003816100381610038162

)

119876 =

1198951

sum119895=1198950

sum119896isinΛ 119895

E(100381610038161003816100381610038161003816119889

1198951198961|119889119895119896|ge120581120582119895

minus 119889119895119896

100381610038161003816100381610038161003816

2

)

119877 =infin

sum119895=1198951+1

sum119896isinΛ 119895

1198892

119895119896

(65)


Let us now investigate sharp upper bounds for 119875 119877 and119876 successively

Upper Bound for 119875 The point (a) of Proposition 10 and2119904(2119904 + 2120588 + 1) lt 1 yield

119875 le 11986221198950

119899le 119862

ln 119899

119899le 119862(

ln 119899

119899)

2119904(2119904+2120588+1)

(66)

Upper Bound for 119877(i) For 119903 ge 1 and119901 ge 2 we have119891 isin 119861119904

119901119903(119872) sube 119861119904

2infin(119872)

Using 2119904(2119904 + 2120588 + 1) lt 2119904(2120588 + 1) we obtain

119877 le 119862infin

sum119895=1198951+1

2minus2119895119904 le 119862((ln 119899)3

119899)

2119904(2120588+1)

le 119862(ln 119899

119899)

2119904(2119904+2120588+1)

(67)

(ii) For 119903 ge 1 and 119901 isin [1 2) we have 119891 isin 119861119904

119901119903(119872) sube

119861119904+12minus1119901

2infin(119872) The condition 119904 gt (2120588 + 1)119901 implies

that (119904 + 12 minus 1119901)(2120588 + 1) gt 119904(2119904 + 2120588 + 1) Thus

119877 le 119862infin

sum119895=1198951+1

2minus2119895(119904+12minus1119901)

le 119862((ln 119899)3

119899)

2(119904+12minus1119901)(2120588+1)

le 119862(ln 119899

119899)

2119904(2119904+2120588+1)

(68)

Hence for 119903 ge 1 119901 ge 2 and 119904 gt 0 or 119901 isin [1 2) and119904 gt (2120588 + 1)119901 we have

119877 le 119862(ln 119899

119899)

2119904(2119904+2120588+1)

(69)

Upper Bound for 119876 Adopting the notation119863119895119896

= 119889119895119896

minus 119889119895119896

119876 can be written as

119876 =4

sum119894=1

119876119894 (70)

where

1198761=

1198951

sum119895=1198950

sum119896isinΛ 119895

E (10038161003816100381610038161003816119863119895119896

100381610038161003816100381610038162

1|119889119895119896|ge120581120582119895 |119889119895119896|lt1205811205821198952

)

1198762=

1198951

sum119895=1198950

sum119896isinΛ 119895

E (10038161003816100381610038161003816119863119895119896

100381610038161003816100381610038162

1|119889119895119896|ge120581120582119895 |119889119895119896|ge1205811205821198952

)

1198763=

1198951

sum119895=1198950

sum119896isinΛ 119895

E (1198892

1198951198961|119889119895119896|lt120581120582119895 |119889119895119896|ge2120581120582119895

)

1198764=

1198951

sum119895=1198950

sum119896isinΛ 119895

E (1198892

1198951198961|119889119895119896|lt120581120582119895 |119889119895119896|lt2120581120582119895

)

(71)

Upper Bound for 1198761

+ 1198763 Owing to the inequalities

1|119889119895119896|lt120581120582119895 |119889119895119896|ge2120581120582119895

le 1|119895119896|gt1205811205821198952

1|119889119895119896|ge120581120582119895 |119889119895119896|lt1205811205821198952

le

1|119895119896|gt1205811205821198952

and 1|119889119895119896|lt120581120582119895 |119889119895119896|ge2120581120582119895

le 1|119889119895119896|le2|119895119896|

theCauchy-Schwarz inequality and the points(b) and (c) ofProposition 10 we have

1198761+ 119876

3le 119862

1198951

sum119895=1198950

sum119896isinΛ 119895

E (10038161003816100381610038161003816119863119895119896

100381610038161003816100381610038162

1|119895119896|gt1205811205821198952

)

le 119862

1198951

sum119895=1198950

sum119896isinΛ 119895

(E (10038161003816100381610038161003816119863119895119896

100381610038161003816100381610038164

))12

(P(10038161003816100381610038161003816119863119895119896

10038161003816100381610038161003816 gt120581120582

119895

2))

12

le 1198621

1198992

1198951

sum119895=1198950

2119895(1+2120588) le 1198621

119899le 119862(

ln 119899

119899)

2119904(2119904+2120588+1)

(72)

Upper Bound for 1198762 It follows from the point(a) of

Proposition 10 that

1198762le

1198951

sum119895=1198950

sum119896isinΛ 119895

E (10038161003816100381610038161003816119863119895119896

100381610038161003816100381610038162

) 1|119889119895119896|ge1205811205821198952

le 1198621

119899

1198951

sum119895=j0

22120588119895 sum119896isinΛ 119895

1|119889119895119896|gt1205811205821198952

(73)

Let us now introduce the integer 119895lowastdefined by

2119895lowast = [(119899

ln 119899)

1(2119904+2120588+1)

] (74)

Note that 119895lowastisin 119895

0 119895

1 for 119899 large enough

Then 1198762can be bounded as

1198762le 119876

21+ 119876

22 (75)

where

11987621

= 1198621

119899

119895lowast

sum119895=1198950

22120588119895 sum119896isinΛ 119895

1|119889119895119896|gt1205811205821198952

11987622

= 1198621

119899

1198951

sum119895=119895lowast+1

22120588119895 sum119896isinΛ 119895

1|119889119895119896|gt1205811205821198952

(76)

On the one hand we have

11987621

le 119862ln 119899

119899

119895lowast

sum119895=1198950

2119895(1+2120588) le 119862(ln 119899

119899)

2119904(2119904+2120588+1)

(77)

On the other hand we have the following


(i) For 119903 ge 1 and 119901 ge 2 the Markov inequality and 119891 isin119861119904

119901119903(119872) sube 119861119904

2infin(119872) yield

11987622

le 119862ln 119899

119899

1198951

sum119895=119895lowast+1

22120588119895 1

1205822

119895

sum119896isinΛ 119895

1198892

119895119896le 119862

infin

sum119895=119895lowast+1

sum119896isinΛ 119895

1198892

119895119896

le 119862infin

sum119895=119895lowast+1

2minus2119895119904 le 119862(ln 119899

119899)

2119904(2119904+2120588+1)

(78)

(ii) For 119903 ge 1 119901 isin [1 2) and 119904 gt (2120588 + 1)119901 the Markovinequality 119891 isin 119861119904

119901119903(119872) and (2119904 + 2120588 + 1)(2 minus 119901)2 +

(119904 + 12 minus 1119901 + 120588 minus 2120588119901)119901 = 2119904 imply that

11987622

le 119862ln 119899

119899

1198951

sum119895=119895lowast+1

22120588119895 1

120582119901

119895

sum119896isinΛ 119895

10038161003816100381610038161003816119889119895119896

10038161003816100381610038161003816119901

le 119862(ln 119899

119899)

(2minus119901)2 infin

sum119895=119895lowast+1

2119895120588(2minus119901)2minus119895(119904+12minus1119901)119901

le 119862(ln 119899

119899)

(2minus119901)2

2minus119895lowast(119904+12minus1119901+120588minus2120588119901)119901

le 119862(ln 119899

119899)

2119904(2119904+2120588+1)

(79)

Therefore for 119903 ge 1 119901 ge 2 and 119904 gt 0 or 119901 isin [1 2) and119904 gt (2120588 + 1)119901 we have

1198762le C(

ln 119899

119899)

2119904(2119904+2120588+1)

(80)

Upper Bound for 1198764 We have

1198764le

1198951

sum119895=1198950

sum119896isinΛ 119895

1198892

1198951198961|119889119895119896|lt2120581120582119895

(81)

Let 119895lowastbe the integer (74) Then 119876

4can be bound as

1198764le 119876

41+ 119876

42 (82)

where

11987641

=

119895lowast

sum119895=1198950

sum119896isinΛ 119895

1198892

1198951198961|119889119895119896|lt2120581120582119895

11987642

=

1198951

sum119895=119895lowast+1

sum119896isinΛ 119895

1198892

1198951198961|119889119895119896|lt2120581120582119895

(83)


11987641

le 119862

119895lowast

sum119895=1198950

21198951205822

119895= 119862

ln 119899

119899

119895lowast

sum119895=1198950

2119895(1+2120588) le 119862(ln 119899

119899)

2119904(2119904+2120588+1)

(84)


(i) For 119903 ge 1 and 119901 ge 2 since 119891 isin 119861119904

119901119903(119872) sube 119861119904

2infin(119872)

we have

11987642

leinfin

sum119895=119895lowast+1

sum119896isinΛ 119895

1198892

119895119896le 119862

infin

sum119895=119895lowast+1

2minus2119895119904 le 119862(ln 119899

119899)

2119904(2119904+2120588+1)

(85)

(ii) For 119903 ge 1 119901 isin [1 2) and 119904 gt (2120588 + 1)119901 owing to theMarkov inequality 119891 isin 119861119904

119901119903(119872) and (2119904 + 2120588 + 1)(2 minus

119901)2 + (119904 + 12 minus 1119901 + 120588 minus 2120588119901)119901 = 2119904 we get

11987642

le 119862

1198951

sum119895=119895lowast+1

1205822minus119901

119895sum

119896isinΛ 119895

10038161003816100381610038161003816119889119895119896

10038161003816100381610038161003816119901

= 119862(ln 119899

119899)

(2minus119901)2 1198951

sum119895=119895lowast+1

2119895120588(2minus119901) sum119896isinΛ 119895

10038161003816100381610038161003816119889119895119896

10038161003816100381610038161003816119901

le 119862(ln 119899

119899)


sum119895=119895lowast+1


le 119862(ln 119899

119899)

(2minus119901)2


le 119862(ln 119899

119899)

2119904(2119904+2120588+1)

(86)

So for 119903 ge 1 119901 ge 2 and 119904 gt 0 or 119901 isin [1 2) and 119904 gt(2120588 + 1)119901 we have

1198764le 119862(

ln 119899

119899)

2119904(2119904+2120588+1)

(87)

Putting (70) (72) (80) and (87) together for 119903 ge 1 119901 ge 2and 119904 gt 0 or 119901 isin [1 2) and 119904 gt (2120588 + 1)119901 we obtain

119876 le 119862(ln 119899

119899)

2119904(2119904+2120588+1)

(88)

Combining (64) (66) (69) and (88) we complete theproof of Theorem 3

Proof of Theorem 4 The proof of Theorem 4 is a direct appli-cation ofTheorem 3 under (G1)ndash(G5) the function 119902 definedby (24) satisfies (H1) see ([42] equation (2)) and (H2) (i) see([42] Lemma 6) (ii) see ([42] equation (11)) and (iii) see([30] Proof of Proposition 61) with 120588 = 120575

Proof of Theorem 5 The proof ofTheorem 5 is a consequenceofTheorem 3 under (J1)ndash(J4) the function 119902 defined by (30)satisfies (H1) and (H2) (i)-(ii) see ([31] Proposition 1) and(iii) see ([52] equation (26)) with 120588 = 120575

Proof of Theorem 6 Set V(119909) = 119891(119909)119892(119909) Following themethodology of [49] we have

119891 (119909) minus 119891 (119909) = 119878 (119909) minus 119879 (119909) (89)


where

119878 (119909)

=1

119892 (119909)(V (119909) minus V (119909)

+119891 (119909) (119892 (119909) minus 119892 (119909))) 1|119892(119909)|ge119888lowast2

119879 (119909) = 119891 (119909) 1|119892(119909)|lt119888lowast2

(90)

Using (K3) and the indicator function we have

|119878 (119909)| le 119862 (|V (119909) minus V (119909)| + 1003816100381610038161003816119892 (119909) minus 119892 (119909)1003816100381610038161003816) (91)

It follows from |119892(119909)| lt 119888lowast2 cap |119892(119909)| gt 119888

lowast sube |119892(119909) minus

119892(119909)| gt 119888lowast2 (K3) (K4) and the Markov inequality that

|119879 (119909)| le 1198621|119892(119909)minus119892(119909)|gt119888lowast2

le 1198621003816100381610038161003816119892 (119909) minus 119892 (119909)

1003816100381610038161003816 (92)

The triangular inequality yields

10038161003816100381610038161003816119891 (119909) minus 119891 (119909)10038161003816100381610038161003816 le 119862 (|V (119909) minus V (119909)| + 1003816100381610038161003816119892 (119909) minus 119892 (119909)

1003816100381610038161003816) (93)

The elementary inequality (119886 + 119887)2 le 2(1198862 + 1198872) implies that

E (10038171003817100381710038171003817119891 minus 119891

100381710038171003817100381710038172

2) le 119862 (E (V minus V2

2) + E (

1003817100381710038171003817119892 minus 11989210038171003817100381710038172

2)) (94)

We now bound this two MISEs via Theorem 3

Upper Bound for the MISE of V Under (K1)ndash(K6) thefunction 119902 defined by (38) satisfies the following

(H1) With V instead of 119891 since 1205851and 119883

1are independent

with E(1205851) = 0

E (119902 (120574119895119896

1198811))

= E (1198841120574119895119896

(1198831)) = E (119891 (119883

1) 120574

119895119896(119883

1))

= int1

0

119891 (119909) 120574119895119896

(119909) 119892 (119909) 119889119909 = int1

0

V (119909) 120574119895119896

(119909) 119889119909

(95)

(H2) (i)-(ii)-(iii) with 120588 = 0

(i) since 1198841(Ω) is bounded thanks to (K2) and (K3) say

|1198841| le 119872 with 119872 gt 0 we have

sup(119909119909lowast)isin1198811(Ω)

10038161003816100381610038161003816119902 (120574119895119896

(119909 119909lowast))

10038161003816100381610038161003816

= sup(119909119909lowast)isin[minus119872119872]times[01]

10038161003816100381610038161003816119909120574119895119896(119909

lowast)10038161003816100381610038161003816

le 119872 sup119909lowastisin[01]

10038161003816100381610038161003816120574119895119896(119909

lowast)10038161003816100381610038161003816 le 11986221198952

(96)

(ii) using the boundedness of 1198841(Ω) then (K4) we have

E (10038161003816100381610038161003816119902(120574119895119896

1198811)100381610038161003816100381610038162

) = E (1198842

1(120574

119895119896(119883

1))

2

)

le 119862E ((120574119895119896

(1198831))

2

)

= 119862int1

0

(120574119895119896

(119909))2

119892 (119909) 119889119909

le 119862int1

0

(120574119895119896

(119909))2

119889119909 le 119862

(97)

(iii) using the boundedness of 1198841(Ω) and making the

change of variables 119910 = 2119895119909 minus 119896 we obtain

int1198811(Ω)

10038161003816100381610038161003816119902 (120574119895119896

119909)10038161003816100381610038161003816 119889119909

= (int119872

minus119872

|119909| 119889119909)(int1

0

10038161003816100381610038161003816120574119895119896(119909

lowast)10038161003816100381610038161003816 119889119909lowast

)

= 1198722 int1

0

10038161003816100381610038161003816120574119895119896(119909)

10038161003816100381610038161003816 119889119909 le 1198622minus1198952

(98)

We conclude by applying Lemma 2 with 120588 = 0 (K5) and (K6)imply (F1) and the previous inequality implies (F2)

Therefore assuming that V isin 119861119904

119901119903(119872) with 119903 ge 1 119901 ge 2

and 119904 isin (0119873) or 119901 isin [1 2) and 119904 isin (1119901119873) Theorem 3proves the existence of a constant 119862 gt 0 satisfying

E (V minus V22) le 119862(

ln 119899

119899)

2119904(2119904+1)

(99)

Upper Bound for theMISE of 119892 Under (K1)ndash(K6) proceedingas the previous point we show that the function 119902 defined by(39) satisfies (H1) with 119892 instead of 119891 and 119883

119905instead of 119881

119905

and(H2) (i)-(ii)-(iii) with 120588 = 0Therefore assuming that 119892 isin 119861119904

119901119903(119872) with 119903 ge 1 119901 ge 2

and 119904 isin (0119873) or 119901 isin [1 2) and 119904 isin (1119901119873) Theorem 3proves the existance of a constant 119862 gt 0 satisfying

E (1003817100381710038171003817119892 minus 119892

10038171003817100381710038172

2) le 119862(

ln 119899

119899)

2119904(2119904+1)

(100)

Combining (94) (99) and (100) we end the proof ofTheorem 6

Conflict of Interests

The author declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The author is thankful to the reviewers for their commentswhich have helped in improving the presentation


References

[1] P DoukhanMixing Properties and Examples vol 85 of LectureNotes in Statistics Springer New York NY USA 1994

[2] M Carrasco and X Chen ldquoMixing and moment properties ofvarious GARCH and stochastic volatility modelsrdquo EconometricTheory vol 18 no 1 pp 17ndash39 2002

[3] R C Bradley Introduction to Strong Mixing Conditions Volume1 2 3 Kendrick Press 2007

[4] P M Robinson ldquoNonparametric estimators for time seriesrdquoJournal of Time Series Analysis vol 4 pp 185ndash207 1983

[5] G G Roussas ldquoNonparametric estimation inmixing sequencesof random variablesrdquo Journal of Statistical Planning and Infer-ence vol 18 no 2 pp 135ndash149 1987

[6] G G Roussas ldquoNonparametric regression estimation undermixing conditionsrdquo Stochastic Processes and their Applicationsvol 36 no 1 pp 107ndash116 1990

[7] Y K Truong and C J Stone ldquoNonparametric function estima-tion involving time seriesrdquo The Annals of Statistics vol 20 pp77ndash97 1992

[8] L H Tran ldquoNonparametric function estimation for time seriesby local average estimatorsrdquoThe Annals of Statistics vol 21 pp1040ndash1057 1993

[9] E Masry ldquoMultivariate local polynomial regression for timeseries uniform strong consistency and ratesrdquo Journal of TimeSeries Analysis vol 17 no 6 pp 571ndash599 1996

[10] E Masry ldquoMultivariate regression estimation local polynomialfitting for time seriesrdquo Stochastic Processes and their Applica-tions vol 65 no 1 pp 81ndash101 1996

[11] E Masry and J Fan ldquoLocal polynomial estimation of regressionfunctions for mixing processesrdquo Scandinavian Journal of Statis-tics vol 24 no 2 pp 165ndash179 1997

[12] D Bosq Nonparametric Statistics for Stochastic Processes Esti-mation and Prediction vol 110 of Lecture Notes in StatisticsSpringer New York NY USA 1998

[13] E Liebscher ldquoEstimation of the density and the regressionfunction under mixing conditionsrdquo Statistics and Decisions vol19 no 1 pp 9ndash26 2001

[14] D L Donoho and I M Johnstone ldquoIdeal spatial adaptation bywavelet shrinkagerdquo Biometrika vol 81 no 3 pp 425ndash455 1994

[15] D L Donoho and I M Johnstone ldquoAdapting to unknownsmoothness via wavelet shrinkagerdquo Journal of the AmericanStatistical Association vol 90 pp 1200ndash1224 1995

[16] D Donoho I Johnstone G Kerkyacharian and D PicardldquoWavelet shrinkage asymptopiardquo Journal of the Royal StatisticalSociety B vol 57 pp 301ndash369 1995

[17] D L Donoho I M Johnstone G Kerkyacharian and DPicard ldquoDensity estimation by wavelet thresholdingrdquo Annals ofStatistics vol 24 no 2 pp 508ndash539 1996

[18] A Antoniadis ldquoWavelets in statistics a review (with discus-sion)rdquo Journal of the Italian Statistical Society vol 6 pp 97ndash1441997

[19] W Hardle G Kerkyacharian D Picard and A TsybakovWavelet Approximation and Statistical Applications vol 129 ofLectures Notes in Statistics Springer New York NY USA 1998

[20] F Leblanc ldquoWavelet linear density estimator for a discrete-timestochastic process 119871

119901-lossesrdquo Statistics and Probability Letters

vol 27 no 1 pp 71ndash84 1996[21] K Tribouley andG Viennet ldquo119871

119901adaptive density estimation in

a alpha-mixing frameworkrdquo Annales de lrsquoinstitut Henri PoincareB vol 34 no 2 pp 179ndash208 1998

[22] E Masry ldquoWavelet-based estimation of multivariate regressionfunctions in besov spacesrdquo Journal of Nonparametric Statisticsvol 12 no 2 pp 283ndash308 2000

[23] P N Patil and Y K Truong ldquoAsymptotics for wavelet basedestimates of piecewise smooth regression for stationary timeseriesrdquo Annals of the Institute of Statistical Mathematics vol 53no 1 pp 159ndash178 2001

[24] H Doosti M Afshari and H A Niroumand ldquoWavelets fornonparametric stochastic regression with mixing stochasticprocessrdquo Communications in Statistics vol 37 no 3 pp 373ndash385 2008

[25] H Doosti and H A Niroumand ldquoMultivariate stochasticregression estimation by wavelet methods for stationary timeseriesrdquo Pakistan Journal of Statistics vol 27 no 1 pp 37ndash462009

[26] H Doosti M S Islam Y P Chaubey and P Gora ldquoTwo-dimensional wavelets for nonlinear autoregressive models withan application in dynamical systemrdquo Italian Journal of Pure andApplied Mathematics no 27 pp 39ndash62 2010

[27] J Cai and H Liang ldquoNonlinear wavelet density estimation fortruncated and dependent observationsrdquo International Journal ofWavelets Multiresolution and Information Processing vol 9 no4 pp 587ndash609 2011

[28] S Niu and H Liang ldquoNonlinear wavelet estimation of condi-tional density under left-truncated and 120572-mixing assumptionsrdquoInternational Journal of Wavelets Multiresolution and Informa-tion Processing vol 9 no 6 pp 989ndash1023 2011

[29] F Benatia and D Yahia ldquoNonlinear wavelet regression functionestimator for censored dependent datardquo Journal Afrika Statis-tika vol 7 no 1 pp 391ndash411 2012

[30] C Chesneau ldquoOn the adaptive wavelet deconvolution of adensity for strong mixing sequencesrdquo Journal of the KoreanStatistical Society vol 41 no 4 pp 423ndash436 2012

[31] C Chesneau ldquoWavelet estimation of a density in a GARCH-type modelrdquo Communications in Statistics vol 42 no 1 pp 98ndash117 2013

[32] C Chesneau ldquoOn the adaptive wavelet estimation of a mul-tidimensional regression function under alpha-mixing depen-dence beyond the standard assumptions on the noiserdquo Com-mentationes Mathematicae Universitatis Carolinae vol 4 pp527ndash556 2013

[33] Y P Chaubey and E Shirazi ldquoOn MISE of a nonlinear waveletestimator of the regression function based on biased data understrong mixing rdquo Communications in Statistics In press

[34] M Abbaszadeh andM Emadi ldquoWavelet density estimation andstatistical evidences role for a garch model in the weighteddistributionrdquo Applied Mathematics vol 4 no 2 pp 10ndash4162013

[35] B Delyon and A Juditsky ldquoOn minimax wavelet estimatorsrdquoApplied and Computational Harmonic Analysis vol 3 no 3 pp215ndash228 1996

[36] G Kerkyacharian and D Picard ldquoThresholding algorithmsmaxisets and well-concentrated basesrdquo Test vol 9 no 2 pp283ndash344 2000

[37] I Daubechies Ten Lectures on Wavelets SIAM 1992[38] A Cohen I Daubechies and P Vial ldquoWavelets on the interval

and fast wavelet transformsrdquo Applied and Computational Har-monic Analysis vol 1 no 1 pp 54ndash81 1993

[39] S Mallat A Wavelet Tour of Signal Processing The sparse waywith contributions fromGabriel Peyralphae ElsevierAcademicPress Amsterdam The Netherlands 3rd edition 2009


[40] Y Meyer Ondelettes et Opalphaerateurs Hermann ParisFrance 1990

[41] V Genon-Catalot T Jeantheau and C Laredo ldquoStochasticvolatility models as hidden Markov models and statisticalapplicationsrdquo Bernoulli vol 6 no 6 pp 1051ndash1079 2000

[42] J Fan and J Koo ldquoWavelet deconvolutionrdquo IEEE Transactionson Information Theory vol 48 no 3 pp 734ndash747 2002

[43] A B Tsybakov Introduction alphaa lrsquoestimation non-paramalphaetrique Springer New York NY USA 2004

[44] E Masry ldquoStrong consistency and rates for deconvolutionof multivariate densities of stationary processesrdquo StochasticProcesses and their Applications vol 47 no 1 pp 53ndash74 1993

[45] R Kulik ldquoNonparametric deconvolution problem for depen-dent sequencesrdquo Electronic Journal of Statistics vol 2 pp 722ndash740 2008

[46] F Comte J Dedecker and M L Taupin ldquoAdaptive densitydeconvolution with dependent inputsrdquo Mathematical Methodsof Statistics vol 17 no 2 pp 87ndash112 2008

[47] H van Zanten and P Zareba ldquoA note on wavelet densitydeconvolution for weakly dependent datardquo Statistical Inferencefor Stochastic Processes vol 11 no 2 pp 207ndash219 2008

[48] W Hardle Applied Nonparametric Regression Cambridge Uni-versity Press Cambridge UK 1990

[49] V A Vasiliev ldquoOne investigation method of a ratios typeestimatorsrdquo in Proceedings of the 16th IFAC Symposium onSystem Identialphacation pp 1ndash6 Brussels Belgium July 2012(in progress)

[50] Y Davydov ldquoThe invariance principle for stationary processesrdquoTheory of Probability and Its Applications vol 15 no 3 pp 498ndash509 1970

[51] C Chesneau ldquoWavelet estimation via block thresholding aminimax study under 119871119901-riskrdquo Statistica Sinica vol 18 no 3pp 1007ndash1024 2008

[52] C Chesneau and H Doosti ldquoWavelet linear density estimationfor a GARCH model under various dependence structuresrdquoJournal of the Iranian Statistical Society vol 11 no 1 pp 1ndash212012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


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OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


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Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


GARCH-type model improving ([30] Proposition 51) and([31] Theorem 2) respectively A last part is devoted to theregression model with random design The obtained resultcompletes the one of Patil and Truong [23]

The organization of this note is as follows In the nextsection we describe the considered wavelet setting Thehard thresholding estimator and its rate of convergenceunder the MISE over Besov balls are presented in Section 3Applications of our general result are given in Section 4 Theproofs are carried out in Section 5

2 Wavelets and Besov Balls

In this section we introduce some notations corresponding towavelets and Besov balls

21 Wavelet Basis We consider the wavelet basis on [0 1]constructs from the Daubechies wavelets db2N with 119873 ge 1(see eg [37]) A brief description of this basis is given belowLet 120601 and 120595 be the initial wavelet functions of the familydb2N These functions have the particularity to be compactlysupported and to belong to the class C119886 for 119873 gt 5119886 For any119895 ge 0 we set Λ

119895= 0 2119895 minus 1 and for 119896 isin Λ

119895

120601119895119896

(119909) = 21198952120601 (2119895119909 minus 119896) 120595119895119896

(119909) = 21198952120595 (2119895119909 minus 119896)

(1)

With appropriated treatments at the boundaries thereexists an integer 120591 such that for any integer ℓ ge 120591 B =120601

ℓ119896 119896 isin Λ

ℓ 120595

119895119896 119895 isin N minus 0 ℓ minus 1 119896 isin Λ

119895 is an

orthonormal basis of L2([0 1]) where

L2([0 1])

= 119891 [0 1] 997888rarr R1003817100381710038171003817119891

10038171003817100381710038172 = (int1

0

1003816100381610038161003816119891 (119909)10038161003816100381610038162

119889119909)

12

lt infin

(2)

For any integer ℓ ge 120591 and 119891 isin L2([0 1]) we have thefollowing wavelet expansion

119891 (119909) = sum119896isinΛ ℓ

119888ℓ119896

120601ℓ119896

(119909) +infin

sum119895=ℓ

sum119896isinΛ 119895

119889119895119896

120595119895119896

(119909)

119909 isin [0 1]

(3)

where 119888119895119896

and 119889119895119896

denote the wavelet coefficients of119891 definedby

119888119895119896

= int1

0

119891 (119909) 120601119895119896

(119909) 119889119909 119889119895119896

= int1

0

119891 (119909) 120595119895119896

(119909) 119889119909

(4)

Technical details can be found in for example Cohen et al[38] and Mallat [39]

In the main result of this paper we will investigate theMISE rate of the proposed estimator by assuming that theunknown function of interest 119891 belongs to a wide class offunctions the Besov class Its definition in terms of waveletcoefficients is presented in the following

22 Besov Balls We say that 119891 isin 119861119904

119901119903(119872)with 119904 gt 0 119901 119903 ge 1

and 119872 gt 0 if and only if there exists a constant 119862 gt 0 suchthat the wavelet coefficients of 119891 given by (4) satisfy

2120591(12minus1119901)( sum119896isinΛ 120591

10038161003816100381610038161198881205911198961003816100381610038161003816119901

)

1119901

+ (infin

sum119895=120591

(2119895(119904+12minus1119901)( sum119896isinΛ 119895

10038161003816100381610038161003816119889119895119896

10038161003816100381610038161003816119901

)

1119901

)

119903

)

1119903

le 119862

(5)

with the usualmodifications if119901 = infin or 119903 = infin Note that forparticular choices of 119904 119901 and 119903119861119904

119901119903(119872) contains the classical

Holder and Sobolev balls (see eg [40] and [19])

Remark 1 We have chosen a wavelet basis on [0 1] to fix thenotations wavelet basis on another interval can be consideredin the rest of the study without affecting the results

3 Statistical Framework Estimator and Result

31 Statistical Framework As mentioned in Section 1 anonparametric estimation setting as general as possible isadopted we aim to estimate an unknown function 119891 isin

L2([0 1]) via 119899 random variables (or vectors) 1198811 119881

119899from

a strictly stationary stochastic process (119881119905)119905isinZ defined on a

probability space (ΩAP) We suppose that (119881119905)119905isinZ has a

120572-mixing dependence structure with exponential decay ratethat is there exist two constants 120574 gt 0 and 120579 gt 0 such that

sup119898ge1

(119890120579119898120572119898) le 120574 (6)

where 120572119898

= sup(119860119861)isinF119881

minusinfin0timesF119881119898infin

|P(119860 cap 119861) minus P(119860)P(119861)|F119881

minusinfin0is the 120590-algebra generated by the random variables (or

vectors) 119881minus1

1198810and F119881

119898infinis the 120590-algebra generated by

the random variables (or vectors) 119881119898 119881

119898+1

The 120572-mixing dependence is reasonably weak it is sat-isfied by a wide variety of models including Markov chainsGARCH-type models and discretely observed diffusions(see for instance [1ndash3 41])

The considered estimator for 119891 is presented below

32 Estimator We define the hard thresholding waveletestimator 119891 by

119891 (119909) = sum119896isinΛ 1198950

1198881198950 119896

1206011198950 119896

(119909) +

1198951

sum119895=1198950

sum119896isinΛ 119895

1198891198951198961|119889119895119896|ge120581120582119895

120595119895119896

(119909)

(7)

where

119888119895119896

=1

119899

119899

sum119894=1

119902 (120601119895119896

119881119894) 119889

119895119896=

1

119899

119899

sum119894=1

119902 (120595119895119896

119881119894) (8)



21198950 = [120591 ln 119899] (9)


satisfying

21198951 = [(119899

(ln 119899)3)

1(2120588+1)

] (10)

120582119895= 2120588119895radic ln 119899

119899 (11)




119895

E (119902 (120574119895119896

1198811)) = int

1

0

119891 (119909) 120574119895119896

(119909) 119889119909 (12)



and 119896 isin Λ119895 (i) sup

119909isin1198811(Ω)|119902(120574

119895119896 119909)| le 119862212058811989521198952 (ii)

E(|119902(120574119895119896


1 ge 110038161003816100381610038161003816C119900V (119902 (120574

119895119896 119881

119898+1) 119902 (120574

119895119896 119881

1))

10038161003816100381610038161003816 le 119862221205881198952minus119895 (13)





119895119896and 119889

119895119896given by




119905)119905isinZ





(1198811 119881119898+1)be the density

of (1198811 119881




sup(119909119910)isin1198811(Ω)times119881119898+1(Ω)

10038161003816100381610038161003816119906(1198811 119881119898+1)(119909 119910)

minus119906 (119909) 119906 (119910)1003816100381610038161003816 le 119862

(14)


0and 119896 isin Λ

119895

int1198811(Ω)

10038161003816100381610038161003816119902 (120574119895119896

119909)10038161003816100381610038161003816 119889119909 le 11986221205881198952minus1198952 (15)






E (10038171003817100381710038171003817119891 minus 119891

100381710038171003817100381710038172

2) le 119862(

ln 119899

119899)

2119904(2119904+2120588+1)

(16)




4 Applications



119881119905= 119883

119905+ 120598

119905 119905 isin Z (17)





119905)119905isinZ is a 120572-mixing


1 119881

119899





sup119909isinR

119891 (119909) le 119862 lt infin (18)



sup119909isinR

119906 (119909) le 119862 (19)



1 119881



sup119898isinZ

sup(119909119910)isinR2

119906(1198811 119881119898+1)

(119909 119910) le 119862 (20)


F (120574) (119909) = intinfin

minusinfin

120574 (119910) 119890minus119894119909119910119889119910 119909 isin R (21)



1003816100381610038161003816F (119892) (119909)1003816100381610038161003816 ge

119888

(1 + 1199092)1205752

(22)


100381610038161003816100381610038161003816(F (119892) (119909))

(ℓ)100381610038161003816100381610038161003816le

119862

(1 + |119909|)120575+ℓ (23)



119902 (120574 119909) =1

2120587int

infin

minusinfin

F (120574) (119910)

F (119892) (119910)119890minus119894119910119909119889119910 (24)





E (10038171003817100381710038171003817119891 minus 119891

100381710038171003817100381710038172

2) le 119862(

ln 119899

119899)

2119904(2119904+2120575+1)

(25)








119881119905= 1205902

119905119885

119905 (26)

(1205902




119905and 119885


119905)119905isinZ is a 120572-


1 119881

119899from (119881



119892 (119909) =1

(120575 minus 1)(minus ln119909)

120575minus1 119909 isin [0 1] (27)


119894=1119880

119894 where

1198801 119880





119906 (119909) le 119862 (28)



1 119881



sup(119909119910)isinR2

119906(1198811 119881119898+1)

(119909 119910) le 119862 (29)



119902 (120574 119909) = 119879120575(120574) (119909) (30)


ℓ(120574)(119909) = 119879(119879





E (10038171003817100381710038171003817119891 minus 119891

100381710038171003817100381710038172

2) le 119862(

ln 119899

119899)

2119904(2119904+2120575+1)

(31)





119905 isin Z 119881119905= (119884

119905 119883

119905)

119884119905= 119891 (119883

119905) + 120585

119905 (32)




119905and 120585




1 119881

119899from (119881





L2([0 1])(K2) 120585

1(Ω) is bounded


sup119909isin[01]

1003816100381610038161003816119891 (119909)1003816100381610038161003816 le 119862 (33)


119888lowastle inf

119909isin[01]

119892 (119909) sup119909isin[01]

119892 (119909) le 119862 (34)




119906 (119909) le 119862 (35)



1 119881



sup119898isinZ


119906(1198811 119881119898+1)

(119909 119910) le 119862 (36)



119891 (119909) =V (119909)119892 (119909)

1|119892(119909)|ge119888lowast2

(37)

where


119902 (120574 (119909 119909lowast)) = 119909120574 (119909

lowast) (38)

and 120588 = 0


119905

119902 (120574 119909) = 120574 (119909) (39)

and 120588 = 0(iii) 119888



119901119903(119872) and 119892 isin 119861119904

119901119903(119872) with 119903 ge 1


E (10038171003817100381710038171003817119891 minus 119891

100381710038171003817100381710038172

2) le 119862(

ln 119899

119899)

2119904(2119904+1)

(40)





5 Proofs







1)|119901) and E(|119896(119882

1)|119902) exist


1) 119896 (119882

119898+1))1003816100381610038161003816

le 1198621205721minus1119901minus1119902

119898(E (

1003816100381610038161003816ℎ (1198821)1003816100381610038161003816119901

))1119901

(E(1003816100381610038161003816119896 (119882

1)1003816100381610038161003816119902

))1119902

(41)






119905= ℎ(119882


1) = 0 and



P(1

119899

100381610038161003816100381610038161003816100381610038161003816

119899

sum119894=1

119880119894

100381610038161003816100381610038161003816100381610038161003816ge 120582)

le 4 exp(minus1205822119899

16 (119863119898119898 + 1205821198721198983)

) + 32119872

120582119899120572

119898

(42)

where

119863119898

= max119897isin12119898

V (119897

sum119894=1

119880119894) (43)



10038161003816100381610038161003816C119900V (119902 (120574119895119896

119881119898+1

) 119902 (120574119895119896

1198811))

10038161003816100381610038161003816

=100381610038161003816100381610038161003816100381610038161003816int1198811(Ω)

int1198811(Ω)

119902 (120574119895119896

119909) 119902 (120574119895119896

119910)

times (119906(1198811 119881119898+1)

(119909 119910) minus 119906 (119909) 119906 (119910)) 119889119909119889119910100381610038161003816100381610038161003816100381610038161003816

le int1198811(Ω)

int1198811(Ω)

10038161003816100381610038161003816119902 (120574119895119896

119909)1003816100381610038161003816100381610038161003816100381610038161003816119902 (120574

119895119896 119910)

10038161003816100381610038161003816

times10038161003816100381610038161003816119906(1198811 119881119898+1)

(119909 119910) minus 119906 (119909) 119906 (119910)10038161003816100381610038161003816 119889119909 119889119910

le 119862(int1198811(Ω)

10038161003816100381610038161003816119902 (120574119895119896

119909)10038161003816100381610038161003816 119889119909)

2

le 119862221205881198952minus119895

(44)




and 120573119895119896


0be (9) and let 119895

1be (10)


0 119895


119895

E (10038161003816100381610038161003816119888119895119896 minus 119888

119895119896

100381610038161003816100381610038162

) le 11986222120588119895 1

119899 (45)

E (10038161003816100381610038161003816119889119895119896

minus 119889119895119896

100381610038161003816100381610038162

) le 11986222120588119895 1

119899 (46)


0 119895


119895

E (10038161003816100381610038161003816119889119895119896

minus 119889119895119896

100381610038161003816100381610038164

) le 11986224120588119895 (47)



1 and


P(10038161003816100381610038161003816119889119895119896

minus 119889119895119896

10038161003816100381610038161003816 ge120581120582

119895

2) le 119862

1

1198994 (48)



E (10038161003816100381610038161003816119888119895119896 minus 119888

119895119896

100381610038161003816100381610038162

) = V (119888119895119896

)

=1

1198992

119899

sum119894=1

V (119902 (120601119895119896

119881119894)) +

2

1198992

times119899

sumV=2

Vminus1

sumℓ=1

Re (C119900V (119902 (120601

119895119896 119881V) 119902 (120601

119895119896 119881

ℓ)))

=1

1198992

119899

sum119894=1

V (119902 (120601119895k 119881119894

)) +2

1198992

times119899minus1

sum119898=1

(119899 minus 119898)Re (C119900V (119902 (120601

119895119896 119881

119898+1) 119902 (120601

119895119896 119881

1)))

le1

119899(E (

10038161003816100381610038161003816119902 (120601119895119896

1198811)100381610038161003816100381610038162

)

+2119899minus1

sum119898=1

10038161003816100381610038161003816C119900V (119902 (120601119895119896

119881119898+1

) 119902 (120601119895119896

1198811))

10038161003816100381610038161003816)

(49)

By (H2)-(ii) we get

E (10038161003816100381610038161003816119902(120601119895119896

1198811)100381610038161003816100381610038162

) le 11986222120588119895 (50)


119899minus1

sum119898=1

10038161003816100381610038161003816C119900V (119902 (120601119895119896

119881119898+1

) 119902 (120601119895119896

1198811))

10038161003816100381610038161003816 = 119860 + 119861 (51)

where

119860 =[ln 119899120579]minus1

sum119898=1

10038161003816100381610038161003816C119900V (119902 (120601119895119896

119881119898+1

) 119902 (120601119895119896

1198811))

10038161003816100381610038161003816

119861 =119899minus1

sum119898=[(ln 119899)120579]

10038161003816100381610038161003816C119900V (119902 (120601119895119896

119881119898+1

) 119902 (120601119895119896

1198811))

10038161003816100381610038161003816

(52)


119860 le 119862221205881198952minus119895 [ln 119899

120579] le 11986222120588119895 (53)




119861 le 119862radicE (10038161003816100381610038161003816119902(120601119895119896

1198811)100381610038161003816100381610038164

)119899minus1

sum119898=[(ln 119899)120579]

radic120572119898

le 119862212058811989521198952radicE (10038161003816100381610038161003816119902(120601119895119896

1198811)100381610038161003816100381610038162

)infin

sum119898=[(ln 119899)120579]

119890minus1205791198982

= 11986222120588119895radic119899119890minus(ln 119899)2 le 11986222120588119895

(54)

Thus119899minus1

sum119898=1

10038161003816100381610038161003816C119900V (119902 (120601119895119896

119881119898+1

) 119902 (120601119895119896

1198811))

10038161003816100381610038161003816 le 11986222120588119895 (55)



| le

sup119909isin1198811(Ω)

|119902(120595119895119896


119895119896| le 119891

2le 119862 that

10038161003816100381610038161003816119889119895119896minus 119889

119895119896

10038161003816100381610038161003816 le10038161003816100381610038161003816119889119895119896

10038161003816100381610038161003816 +10038161003816100381610038161003816119889119895119896

10038161003816100381610038161003816 le 119862212058811989521198952 (56)


E (10038161003816100381610038161003816119889119895119896

minus 119889119895119896

100381610038161003816100381610038164

) le 119862221205881198952119895E (

10038161003816100381610038161003816119889119895119896minus 119889

119895119896

100381610038161003816100381610038162

) le 119862241205881198952119895 1

119899

(57)


Lemma 9 Let us set

119880119894= 119902 (120595

119895119896 119881

119894) minus E (119902 (120595

119895119896 119881

1)) (58)


119899(ln 119899)3

1003816100381610038161003816119880119894

1003816100381610038161003816 le 2 sup119909isin1198811(Ω)

10038161003816100381610038161003816119902 (120595119895119896

119909)10038161003816100381610038161003816 le 119862212058811989521198952 le 1198622120588119895

radic119899

(ln 119899)3

(59)



V (119897

sum119894=1

119880119894)

= V (119897

sum119894=1

119902 (120595119895119896

119881119894)) le 11986222120588119895 (119897 + 11989722minus119895) le 11986222120588119895119897

(60)

Therefore

119863119898

= max119897isin12119898

V (119897

sum119894=1

119880119894) le 11986222120588119895119898 (61)


119899 120582 = 120581120582

1198952


P(10038161003816100381610038161003816119889119895119896

minus 119889119895119896

10038161003816100381610038161003816 ge120581120582

119895

2)

le 119862(exp(minus11986212058121205822

119895119899

119863119898119898 + 120581120582

119895119898119872

) +119872

120582119895

119899119890minus120579119898)

le 119862( exp (minus119862((120581222120588119895 ln 119899)

times (22120588119895 + 1205812120588119895radic (ln 119899)

119899

times [radic120581 ln 119899] 2120588119895

radic119899

(ln 119899)3)

minus1

))

+ radic119899(ln 119899)3

(ln 119899) 119899119899119890minus120579[radic120581 ln 119899])

le 119862(119899minus1198621205812(1+120581

32) + 1198992minus120579radic120581)

(62)





119891 (119909) = sum119896isinΛ 1198950

1198881198950 119896

1206011198950 119896

(119909) +infin

sum119895=1198950

sum119896isinΛ 119895

119889119895119896

120595119895119896

(119909) (63)

where 1198881198950 119896

= int1

0119891(119909)120601

1198950 119896(119909)119889119909 and 119889

119895119896= int

1

0119891(119909)120595

119895119896(119909)119889119909


E (10038171003817100381710038171003817119891 minus 119891

100381710038171003817100381710038172

2) = 119875 + 119876 + 119877 (64)

where

119875 = sum119896isinΛ 1198950

E (100381610038161003816100381610038161198881198950 119896 minus 119888

1198950 119896

100381610038161003816100381610038162

)

119876 =

1198951

sum119895=1198950

sum119896isinΛ 119895

E(100381610038161003816100381610038161003816119889

1198951198961|119889119895119896|ge120581120582119895

minus 119889119895119896

100381610038161003816100381610038161003816

2

)

119877 =infin

sum119895=1198951+1

sum119896isinΛ 119895

1198892

119895119896

(65)




119875 le 11986221198950

119899le 119862

ln 119899

119899le 119862(

ln 119899

119899)

2119904(2119904+2120588+1)

(66)


119901119903(119872) sube 119861119904

2infin(119872)


119877 le 119862infin

sum119895=1198951+1

2minus2119895119904 le 119862((ln 119899)3

119899)

2119904(2120588+1)

le 119862(ln 119899

119899)

2119904(2119904+2120588+1)

(67)


119901119903(119872) sube

119861119904+12minus1119901



119877 le 119862infin

sum119895=1198951+1

2minus2119895(119904+12minus1119901)

le 119862((ln 119899)3

119899)

2(119904+12minus1119901)(2120588+1)

le 119862(ln 119899

119899)

2119904(2119904+2120588+1)

(68)


119877 le 119862(ln 119899

119899)

2119904(2119904+2120588+1)

(69)


= 119889119895119896

minus 119889119895119896


119876 =4

sum119894=1

119876119894 (70)

where

1198761=

1198951

sum119895=1198950

sum119896isinΛ 119895

E (10038161003816100381610038161003816119863119895119896

100381610038161003816100381610038162

1|119889119895119896|ge120581120582119895 |119889119895119896|lt1205811205821198952

)

1198762=

1198951

sum119895=1198950

sum119896isinΛ 119895

E (10038161003816100381610038161003816119863119895119896

100381610038161003816100381610038162

1|119889119895119896|ge120581120582119895 |119889119895119896|ge1205811205821198952

)

1198763=

1198951

sum119895=1198950

sum119896isinΛ 119895

E (1198892

1198951198961|119889119895119896|lt120581120582119895 |119889119895119896|ge2120581120582119895

)

1198764=

1198951

sum119895=1198950

sum119896isinΛ 119895

E (1198892

1198951198961|119889119895119896|lt120581120582119895 |119889119895119896|lt2120581120582119895

)

(71)



1|119889119895119896|lt120581120582119895 |119889119895119896|ge2120581120582119895

le 1|119895119896|gt1205811205821198952

1|119889119895119896|ge120581120582119895 |119889119895119896|lt1205811205821198952

le

1|119895119896|gt1205811205821198952

and 1|119889119895119896|lt120581120582119895 |119889119895119896|ge2120581120582119895

le 1|119889119895119896|le2|119895119896|


1198761+ 119876

3le 119862

1198951

sum119895=1198950

sum119896isinΛ 119895

E (10038161003816100381610038161003816119863119895119896

100381610038161003816100381610038162

1|119895119896|gt1205811205821198952

)

le 119862

1198951

sum119895=1198950

sum119896isinΛ 119895

(E (10038161003816100381610038161003816119863119895119896

100381610038161003816100381610038164

))12

(P(10038161003816100381610038161003816119863119895119896

10038161003816100381610038161003816 gt120581120582

119895

2))

12

le 1198621

1198992

1198951

sum119895=1198950

2119895(1+2120588) le 1198621

119899le 119862(

ln 119899

119899)

2119904(2119904+2120588+1)

(72)


Proposition 10 that

1198762le

1198951

sum119895=1198950

sum119896isinΛ 119895

E (10038161003816100381610038161003816119863119895119896

100381610038161003816100381610038162

) 1|119889119895119896|ge1205811205821198952

le 1198621

119899

1198951

sum119895=j0

22120588119895 sum119896isinΛ 119895

1|119889119895119896|gt1205811205821198952

(73)


2119895lowast = [(119899

ln 119899)

1(2119904+2120588+1)

] (74)


0 119895



1198762le 119876

21+ 119876

22 (75)

where

11987621

= 1198621

119899

119895lowast

sum119895=1198950

22120588119895 sum119896isinΛ 119895

1|119889119895119896|gt1205811205821198952

11987622

= 1198621

119899

1198951

sum119895=119895lowast+1

22120588119895 sum119896isinΛ 119895

1|119889119895119896|gt1205811205821198952

(76)


11987621

le 119862ln 119899

119899

119895lowast

sum119895=1198950

2119895(1+2120588) le 119862(ln 119899

119899)

2119904(2119904+2120588+1)

(77)




119901119903(119872) sube 119861119904


11987622

le 119862ln 119899

119899

1198951

sum119895=119895lowast+1

22120588119895 1

1205822

119895

sum119896isinΛ 119895

1198892

119895119896le 119862

infin

sum119895=119895lowast+1

sum119896isinΛ 119895

1198892

119895119896

le 119862infin

sum119895=119895lowast+1

2minus2119895119904 le 119862(ln 119899

119899)

2119904(2119904+2120588+1)

(78)


119901119903(119872) and (2119904 + 2120588 + 1)(2 minus 119901)2 +


11987622

le 119862ln 119899

119899

1198951

sum119895=119895lowast+1

22120588119895 1

120582119901

119895

sum119896isinΛ 119895

10038161003816100381610038161003816119889119895119896

10038161003816100381610038161003816119901

le 119862(ln 119899

119899)


sum119895=119895lowast+1


le 119862(ln 119899

119899)

(2minus119901)2


le 119862(ln 119899

119899)

2119904(2119904+2120588+1)

(79)


1198762le C(

ln 119899

119899)

2119904(2119904+2120588+1)

(80)


1198764le

1198951

sum119895=1198950

sum119896isinΛ 119895

1198892

1198951198961|119889119895119896|lt2120581120582119895

(81)


4can be bound as

1198764le 119876

41+ 119876

42 (82)

where

11987641

=

119895lowast

sum119895=1198950

sum119896isinΛ 119895

1198892

1198951198961|119889119895119896|lt2120581120582119895

11987642

=

1198951

sum119895=119895lowast+1

sum119896isinΛ 119895

1198892

1198951198961|119889119895119896|lt2120581120582119895

(83)


11987641

le 119862

119895lowast

sum119895=1198950

21198951205822

119895= 119862

ln 119899

119899

119895lowast

sum119895=1198950

2119895(1+2120588) le 119862(ln 119899

119899)

2119904(2119904+2120588+1)

(84)



119901119903(119872) sube 119861119904

2infin(119872)

we have

11987642

leinfin

sum119895=119895lowast+1

sum119896isinΛ 119895

1198892

119895119896le 119862

infin

sum119895=119895lowast+1

2minus2119895119904 le 119862(ln 119899

119899)

2119904(2119904+2120588+1)

(85)


119901119903(119872) and (2119904 + 2120588 + 1)(2 minus


11987642

le 119862

1198951

sum119895=119895lowast+1

1205822minus119901

119895sum

119896isinΛ 119895

10038161003816100381610038161003816119889119895119896

10038161003816100381610038161003816119901

= 119862(ln 119899

119899)

(2minus119901)2 1198951

sum119895=119895lowast+1

2119895120588(2minus119901) sum119896isinΛ 119895

10038161003816100381610038161003816119889119895119896

10038161003816100381610038161003816119901

le 119862(ln 119899

119899)


sum119895=119895lowast+1


le 119862(ln 119899

119899)

(2minus119901)2


le 119862(ln 119899

119899)

2119904(2119904+2120588+1)

(86)


1198764le 119862(

ln 119899

119899)

2119904(2119904+2120588+1)

(87)


119876 le 119862(ln 119899

119899)

2119904(2119904+2120588+1)

(88)





119891 (119909) minus 119891 (119909) = 119878 (119909) minus 119879 (119909) (89)


where

119878 (119909)

=1

119892 (119909)(V (119909) minus V (119909)

+119891 (119909) (119892 (119909) minus 119892 (119909))) 1|119892(119909)|ge119888lowast2

119879 (119909) = 119891 (119909) 1|119892(119909)|lt119888lowast2

(90)


|119878 (119909)| le 119862 (|V (119909) minus V (119909)| + 1003816100381610038161003816119892 (119909) minus 119892 (119909)1003816100381610038161003816) (91)





le 1198621003816100381610038161003816119892 (119909) minus 119892 (119909)

1003816100381610038161003816 (92)



1003816100381610038161003816) (93)


E (10038171003817100381710038171003817119891 minus 119891

100381710038171003817100381710038172

2) le 119862 (E (V minus V2

2) + E (

1003817100381710038171003817119892 minus 11989210038171003817100381710038172

2)) (94)




1are independent

with E(1205851) = 0

E (119902 (120574119895119896

1198811))

= E (1198841120574119895119896

(1198831)) = E (119891 (119883

1) 120574

119895119896(119883

1))

= int1

0

119891 (119909) 120574119895119896

(119909) 119892 (119909) 119889119909 = int1

0

V (119909) 120574119895119896

(119909) 119889119909

(95)

(H2) (i)-(ii)-(iii) with 120588 = 0


|1198841| le 119872 with 119872 gt 0 we have

sup(119909119909lowast)isin1198811(Ω)

10038161003816100381610038161003816119902 (120574119895119896

(119909 119909lowast))

10038161003816100381610038161003816


10038161003816100381610038161003816119909120574119895119896(119909

lowast)10038161003816100381610038161003816


10038161003816100381610038161003816120574119895119896(119909

lowast)10038161003816100381610038161003816 le 11986221198952

(96)


E (10038161003816100381610038161003816119902(120574119895119896

1198811)100381610038161003816100381610038162

) = E (1198842

1(120574

119895119896(119883

1))

2

)

le 119862E ((120574119895119896

(1198831))

2

)

= 119862int1

0

(120574119895119896

(119909))2

119892 (119909) 119889119909

le 119862int1

0

(120574119895119896

(119909))2

119889119909 le 119862

(97)



int1198811(Ω)

10038161003816100381610038161003816119902 (120574119895119896

119909)10038161003816100381610038161003816 119889119909

= (int119872

minus119872

|119909| 119889119909)(int1

0

10038161003816100381610038161003816120574119895119896(119909

lowast)10038161003816100381610038161003816 119889119909lowast

)

= 1198722 int1

0

10038161003816100381610038161003816120574119895119896(119909)

10038161003816100381610038161003816 119889119909 le 1198622minus1198952

(98)



119901119903(119872) with 119903 ge 1 119901 ge 2


E (V minus V22) le 119862(

ln 119899

119899)

2119904(2119904+1)

(99)


119905instead of 119881

119905


119901119903(119872) with 119903 ge 1 119901 ge 2


E (1003817100381710038171003817119892 minus 119892

10038171003817100381710038172

2) le 119862(

ln 119899

119899)

2119904(2119904+1)

(100)




Acknowledgment



References
































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











21198950 = [120591 ln 119899] (9)


satisfying

21198951 = [(119899

(ln 119899)3)

1(2120588+1)

] (10)

120582119895= 2120588119895radic ln 119899

119899 (11)




119895

E (119902 (120574119895119896

1198811)) = int

1

0

119891 (119909) 120574119895119896

(119909) 119889119909 (12)



and 119896 isin Λ119895 (i) sup

119909isin1198811(Ω)|119902(120574

119895119896 119909)| le 119862212058811989521198952 (ii)

E(|119902(120574119895119896


1 ge 110038161003816100381610038161003816C119900V (119902 (120574

119895119896 119881

119898+1) 119902 (120574

119895119896 119881

1))

10038161003816100381610038161003816 le 119862221205881198952minus119895 (13)





119895119896and 119889

119895119896given by




119905)119905isinZ





(1198811 119881119898+1)be the density

of (1198811 119881




sup(119909119910)isin1198811(Ω)times119881119898+1(Ω)

10038161003816100381610038161003816119906(1198811 119881119898+1)(119909 119910)

minus119906 (119909) 119906 (119910)1003816100381610038161003816 le 119862

(14)


0and 119896 isin Λ

119895

int1198811(Ω)

10038161003816100381610038161003816119902 (120574119895119896

119909)10038161003816100381610038161003816 119889119909 le 11986221205881198952minus1198952 (15)






E (10038171003817100381710038171003817119891 minus 119891

100381710038171003817100381710038172

2) le 119862(

ln 119899

119899)

2119904(2119904+2120588+1)

(16)




4 Applications



119881119905= 119883

119905+ 120598

119905 119905 isin Z (17)





119905)119905isinZ is a 120572-mixing


1 119881

119899





sup119909isinR

119891 (119909) le 119862 lt infin (18)



sup119909isinR

119906 (119909) le 119862 (19)



1 119881



sup119898isinZ

sup(119909119910)isinR2

119906(1198811 119881119898+1)

(119909 119910) le 119862 (20)


F (120574) (119909) = intinfin

minusinfin

120574 (119910) 119890minus119894119909119910119889119910 119909 isin R (21)



1003816100381610038161003816F (119892) (119909)1003816100381610038161003816 ge

119888

(1 + 1199092)1205752

(22)


100381610038161003816100381610038161003816(F (119892) (119909))

(ℓ)100381610038161003816100381610038161003816le

119862

(1 + |119909|)120575+ℓ (23)



119902 (120574 119909) =1

2120587int

infin

minusinfin

F (120574) (119910)

F (119892) (119910)119890minus119894119910119909119889119910 (24)





E (10038171003817100381710038171003817119891 minus 119891

100381710038171003817100381710038172

2) le 119862(

ln 119899

119899)

2119904(2119904+2120575+1)

(25)








119881119905= 1205902

119905119885

119905 (26)

(1205902




119905and 119885


119905)119905isinZ is a 120572-


1 119881

119899from (119881



119892 (119909) =1

(120575 minus 1)(minus ln119909)

120575minus1 119909 isin [0 1] (27)


119894=1119880

119894 where

1198801 119880





119906 (119909) le 119862 (28)



1 119881



sup(119909119910)isinR2

119906(1198811 119881119898+1)

(119909 119910) le 119862 (29)



119902 (120574 119909) = 119879120575(120574) (119909) (30)


ℓ(120574)(119909) = 119879(119879





E (10038171003817100381710038171003817119891 minus 119891

100381710038171003817100381710038172

2) le 119862(

ln 119899

119899)

2119904(2119904+2120575+1)

(31)





119905 isin Z 119881119905= (119884

119905 119883

119905)

119884119905= 119891 (119883

119905) + 120585

119905 (32)




119905and 120585




1 119881

119899from (119881





L2([0 1])(K2) 120585

1(Ω) is bounded


sup119909isin[01]

1003816100381610038161003816119891 (119909)1003816100381610038161003816 le 119862 (33)


119888lowastle inf

119909isin[01]

119892 (119909) sup119909isin[01]

119892 (119909) le 119862 (34)




119906 (119909) le 119862 (35)



1 119881



sup119898isinZ


119906(1198811 119881119898+1)

(119909 119910) le 119862 (36)



119891 (119909) =V (119909)119892 (119909)

1|119892(119909)|ge119888lowast2

(37)

where


119902 (120574 (119909 119909lowast)) = 119909120574 (119909

lowast) (38)

and 120588 = 0


119905

119902 (120574 119909) = 120574 (119909) (39)

and 120588 = 0(iii) 119888



119901119903(119872) and 119892 isin 119861119904

119901119903(119872) with 119903 ge 1


E (10038171003817100381710038171003817119891 minus 119891

100381710038171003817100381710038172

2) le 119862(

ln 119899

119899)

2119904(2119904+1)

(40)





5 Proofs







1)|119901) and E(|119896(119882

1)|119902) exist


1) 119896 (119882

119898+1))1003816100381610038161003816

le 1198621205721minus1119901minus1119902

119898(E (

1003816100381610038161003816ℎ (1198821)1003816100381610038161003816119901

))1119901

(E(1003816100381610038161003816119896 (119882

1)1003816100381610038161003816119902

))1119902

(41)






119905= ℎ(119882


1) = 0 and



P(1

119899

100381610038161003816100381610038161003816100381610038161003816

119899

sum119894=1

119880119894

100381610038161003816100381610038161003816100381610038161003816ge 120582)

le 4 exp(minus1205822119899

16 (119863119898119898 + 1205821198721198983)

) + 32119872

120582119899120572

119898

(42)

where

119863119898

= max119897isin12119898

V (119897

sum119894=1

119880119894) (43)



10038161003816100381610038161003816C119900V (119902 (120574119895119896

119881119898+1

) 119902 (120574119895119896

1198811))

10038161003816100381610038161003816

=100381610038161003816100381610038161003816100381610038161003816int1198811(Ω)

int1198811(Ω)

119902 (120574119895119896

119909) 119902 (120574119895119896

119910)

times (119906(1198811 119881119898+1)

(119909 119910) minus 119906 (119909) 119906 (119910)) 119889119909119889119910100381610038161003816100381610038161003816100381610038161003816

le int1198811(Ω)

int1198811(Ω)

10038161003816100381610038161003816119902 (120574119895119896

119909)1003816100381610038161003816100381610038161003816100381610038161003816119902 (120574

119895119896 119910)

10038161003816100381610038161003816

times10038161003816100381610038161003816119906(1198811 119881119898+1)

(119909 119910) minus 119906 (119909) 119906 (119910)10038161003816100381610038161003816 119889119909 119889119910

le 119862(int1198811(Ω)

10038161003816100381610038161003816119902 (120574119895119896

119909)10038161003816100381610038161003816 119889119909)

2

le 119862221205881198952minus119895

(44)




and 120573119895119896


0be (9) and let 119895

1be (10)


0 119895


119895

E (10038161003816100381610038161003816119888119895119896 minus 119888

119895119896

100381610038161003816100381610038162

) le 11986222120588119895 1

119899 (45)

E (10038161003816100381610038161003816119889119895119896

minus 119889119895119896

100381610038161003816100381610038162

) le 11986222120588119895 1

119899 (46)


0 119895


119895

E (10038161003816100381610038161003816119889119895119896

minus 119889119895119896

100381610038161003816100381610038164

) le 11986224120588119895 (47)



1 and


P(10038161003816100381610038161003816119889119895119896

minus 119889119895119896

10038161003816100381610038161003816 ge120581120582

119895

2) le 119862

1

1198994 (48)



E (10038161003816100381610038161003816119888119895119896 minus 119888

119895119896

100381610038161003816100381610038162

) = V (119888119895119896

)

=1

1198992

119899

sum119894=1

V (119902 (120601119895119896

119881119894)) +

2

1198992

times119899

sumV=2

Vminus1

sumℓ=1

Re (C119900V (119902 (120601

119895119896 119881V) 119902 (120601

119895119896 119881

ℓ)))

=1

1198992

119899

sum119894=1

V (119902 (120601119895k 119881119894

)) +2

1198992

times119899minus1

sum119898=1

(119899 minus 119898)Re (C119900V (119902 (120601

119895119896 119881

119898+1) 119902 (120601

119895119896 119881

1)))

le1

119899(E (

10038161003816100381610038161003816119902 (120601119895119896

1198811)100381610038161003816100381610038162

)

+2119899minus1

sum119898=1

10038161003816100381610038161003816C119900V (119902 (120601119895119896

119881119898+1

) 119902 (120601119895119896

1198811))

10038161003816100381610038161003816)

(49)

By (H2)-(ii) we get

E (10038161003816100381610038161003816119902(120601119895119896

1198811)100381610038161003816100381610038162

) le 11986222120588119895 (50)


119899minus1

sum119898=1

10038161003816100381610038161003816C119900V (119902 (120601119895119896

119881119898+1

) 119902 (120601119895119896

1198811))

10038161003816100381610038161003816 = 119860 + 119861 (51)

where

119860 =[ln 119899120579]minus1

sum119898=1

10038161003816100381610038161003816C119900V (119902 (120601119895119896

119881119898+1

) 119902 (120601119895119896

1198811))

10038161003816100381610038161003816

119861 =119899minus1

sum119898=[(ln 119899)120579]

10038161003816100381610038161003816C119900V (119902 (120601119895119896

119881119898+1

) 119902 (120601119895119896

1198811))

10038161003816100381610038161003816

(52)


119860 le 119862221205881198952minus119895 [ln 119899

120579] le 11986222120588119895 (53)




119861 le 119862radicE (10038161003816100381610038161003816119902(120601119895119896

1198811)100381610038161003816100381610038164

)119899minus1

sum119898=[(ln 119899)120579]

radic120572119898

le 119862212058811989521198952radicE (10038161003816100381610038161003816119902(120601119895119896

1198811)100381610038161003816100381610038162

)infin

sum119898=[(ln 119899)120579]

119890minus1205791198982

= 11986222120588119895radic119899119890minus(ln 119899)2 le 11986222120588119895

(54)

Thus119899minus1

sum119898=1

10038161003816100381610038161003816C119900V (119902 (120601119895119896

119881119898+1

) 119902 (120601119895119896

1198811))

10038161003816100381610038161003816 le 11986222120588119895 (55)



| le

sup119909isin1198811(Ω)

|119902(120595119895119896


119895119896| le 119891

2le 119862 that

10038161003816100381610038161003816119889119895119896minus 119889

119895119896

10038161003816100381610038161003816 le10038161003816100381610038161003816119889119895119896

10038161003816100381610038161003816 +10038161003816100381610038161003816119889119895119896

10038161003816100381610038161003816 le 119862212058811989521198952 (56)


E (10038161003816100381610038161003816119889119895119896

minus 119889119895119896

100381610038161003816100381610038164

) le 119862221205881198952119895E (

10038161003816100381610038161003816119889119895119896minus 119889

119895119896

100381610038161003816100381610038162

) le 119862241205881198952119895 1

119899

(57)


Lemma 9 Let us set

119880119894= 119902 (120595

119895119896 119881

119894) minus E (119902 (120595

119895119896 119881

1)) (58)


119899(ln 119899)3

1003816100381610038161003816119880119894

1003816100381610038161003816 le 2 sup119909isin1198811(Ω)

10038161003816100381610038161003816119902 (120595119895119896

119909)10038161003816100381610038161003816 le 119862212058811989521198952 le 1198622120588119895

radic119899

(ln 119899)3

(59)



V (119897

sum119894=1

119880119894)

= V (119897

sum119894=1

119902 (120595119895119896

119881119894)) le 11986222120588119895 (119897 + 11989722minus119895) le 11986222120588119895119897

(60)

Therefore

119863119898

= max119897isin12119898

V (119897

sum119894=1

119880119894) le 11986222120588119895119898 (61)


119899 120582 = 120581120582

1198952


P(10038161003816100381610038161003816119889119895119896

minus 119889119895119896

10038161003816100381610038161003816 ge120581120582

119895

2)

le 119862(exp(minus11986212058121205822

119895119899

119863119898119898 + 120581120582

119895119898119872

) +119872

120582119895

119899119890minus120579119898)

le 119862( exp (minus119862((120581222120588119895 ln 119899)

times (22120588119895 + 1205812120588119895radic (ln 119899)

119899

times [radic120581 ln 119899] 2120588119895

radic119899

(ln 119899)3)

minus1

))

+ radic119899(ln 119899)3

(ln 119899) 119899119899119890minus120579[radic120581 ln 119899])

le 119862(119899minus1198621205812(1+120581

32) + 1198992minus120579radic120581)

(62)





119891 (119909) = sum119896isinΛ 1198950

1198881198950 119896

1206011198950 119896

(119909) +infin

sum119895=1198950

sum119896isinΛ 119895

119889119895119896

120595119895119896

(119909) (63)

where 1198881198950 119896

= int1

0119891(119909)120601

1198950 119896(119909)119889119909 and 119889

119895119896= int

1

0119891(119909)120595

119895119896(119909)119889119909


E (10038171003817100381710038171003817119891 minus 119891

100381710038171003817100381710038172

2) = 119875 + 119876 + 119877 (64)

where

119875 = sum119896isinΛ 1198950

E (100381610038161003816100381610038161198881198950 119896 minus 119888

1198950 119896

100381610038161003816100381610038162

)

119876 =

1198951

sum119895=1198950

sum119896isinΛ 119895

E(100381610038161003816100381610038161003816119889

1198951198961|119889119895119896|ge120581120582119895

minus 119889119895119896

100381610038161003816100381610038161003816

2

)

119877 =infin

sum119895=1198951+1

sum119896isinΛ 119895

1198892

119895119896

(65)




119875 le 11986221198950

119899le 119862

ln 119899

119899le 119862(

ln 119899

119899)

2119904(2119904+2120588+1)

(66)


119901119903(119872) sube 119861119904

2infin(119872)


119877 le 119862infin

sum119895=1198951+1

2minus2119895119904 le 119862((ln 119899)3

119899)

2119904(2120588+1)

le 119862(ln 119899

119899)

2119904(2119904+2120588+1)

(67)


119901119903(119872) sube

119861119904+12minus1119901



119877 le 119862infin

sum119895=1198951+1

2minus2119895(119904+12minus1119901)

le 119862((ln 119899)3

119899)

2(119904+12minus1119901)(2120588+1)

le 119862(ln 119899

119899)

2119904(2119904+2120588+1)

(68)


119877 le 119862(ln 119899

119899)

2119904(2119904+2120588+1)

(69)


= 119889119895119896

minus 119889119895119896


119876 =4

sum119894=1

119876119894 (70)

where

1198761=

1198951

sum119895=1198950

sum119896isinΛ 119895

E (10038161003816100381610038161003816119863119895119896

100381610038161003816100381610038162

1|119889119895119896|ge120581120582119895 |119889119895119896|lt1205811205821198952

)

1198762=

1198951

sum119895=1198950

sum119896isinΛ 119895

E (10038161003816100381610038161003816119863119895119896

100381610038161003816100381610038162

1|119889119895119896|ge120581120582119895 |119889119895119896|ge1205811205821198952

)

1198763=

1198951

sum119895=1198950

sum119896isinΛ 119895

E (1198892

1198951198961|119889119895119896|lt120581120582119895 |119889119895119896|ge2120581120582119895

)

1198764=

1198951

sum119895=1198950

sum119896isinΛ 119895

E (1198892

1198951198961|119889119895119896|lt120581120582119895 |119889119895119896|lt2120581120582119895

)

(71)



1|119889119895119896|lt120581120582119895 |119889119895119896|ge2120581120582119895

le 1|119895119896|gt1205811205821198952

1|119889119895119896|ge120581120582119895 |119889119895119896|lt1205811205821198952

le

1|119895119896|gt1205811205821198952

and 1|119889119895119896|lt120581120582119895 |119889119895119896|ge2120581120582119895

le 1|119889119895119896|le2|119895119896|


1198761+ 119876

3le 119862

1198951

sum119895=1198950

sum119896isinΛ 119895

E (10038161003816100381610038161003816119863119895119896

100381610038161003816100381610038162

1|119895119896|gt1205811205821198952

)

le 119862

1198951

sum119895=1198950

sum119896isinΛ 119895

(E (10038161003816100381610038161003816119863119895119896

100381610038161003816100381610038164

))12

(P(10038161003816100381610038161003816119863119895119896

10038161003816100381610038161003816 gt120581120582

119895

2))

12

le 1198621

1198992

1198951

sum119895=1198950

2119895(1+2120588) le 1198621

119899le 119862(

ln 119899

119899)

2119904(2119904+2120588+1)

(72)


Proposition 10 that

1198762le

1198951

sum119895=1198950

sum119896isinΛ 119895

E (10038161003816100381610038161003816119863119895119896

100381610038161003816100381610038162

) 1|119889119895119896|ge1205811205821198952

le 1198621

119899

1198951

sum119895=j0

22120588119895 sum119896isinΛ 119895

1|119889119895119896|gt1205811205821198952

(73)


2119895lowast = [(119899

ln 119899)

1(2119904+2120588+1)

] (74)


0 119895



1198762le 119876

21+ 119876

22 (75)

where

11987621

= 1198621

119899

119895lowast

sum119895=1198950

22120588119895 sum119896isinΛ 119895

1|119889119895119896|gt1205811205821198952

11987622

= 1198621

119899

1198951

sum119895=119895lowast+1

22120588119895 sum119896isinΛ 119895

1|119889119895119896|gt1205811205821198952

(76)


11987621

le 119862ln 119899

119899

119895lowast

sum119895=1198950

2119895(1+2120588) le 119862(ln 119899

119899)

2119904(2119904+2120588+1)

(77)




119901119903(119872) sube 119861119904


11987622

le 119862ln 119899

119899

1198951

sum119895=119895lowast+1

22120588119895 1

1205822

119895

sum119896isinΛ 119895

1198892

119895119896le 119862

infin

sum119895=119895lowast+1

sum119896isinΛ 119895

1198892

119895119896

le 119862infin

sum119895=119895lowast+1

2minus2119895119904 le 119862(ln 119899

119899)

2119904(2119904+2120588+1)

(78)


119901119903(119872) and (2119904 + 2120588 + 1)(2 minus 119901)2 +


11987622

le 119862ln 119899

119899

1198951

sum119895=119895lowast+1

22120588119895 1

120582119901

119895

sum119896isinΛ 119895

10038161003816100381610038161003816119889119895119896

10038161003816100381610038161003816119901

le 119862(ln 119899

119899)


sum119895=119895lowast+1


le 119862(ln 119899

119899)

(2minus119901)2


le 119862(ln 119899

119899)

2119904(2119904+2120588+1)

(79)


1198762le C(

ln 119899

119899)

2119904(2119904+2120588+1)

(80)


1198764le

1198951

sum119895=1198950

sum119896isinΛ 119895

1198892

1198951198961|119889119895119896|lt2120581120582119895

(81)


4can be bound as

1198764le 119876

41+ 119876

42 (82)

where

11987641

=

119895lowast

sum119895=1198950

sum119896isinΛ 119895

1198892

1198951198961|119889119895119896|lt2120581120582119895

11987642

=

1198951

sum119895=119895lowast+1

sum119896isinΛ 119895

1198892

1198951198961|119889119895119896|lt2120581120582119895

(83)


11987641

le 119862

119895lowast

sum119895=1198950

21198951205822

119895= 119862

ln 119899

119899

119895lowast

sum119895=1198950

2119895(1+2120588) le 119862(ln 119899

119899)

2119904(2119904+2120588+1)

(84)



119901119903(119872) sube 119861119904

2infin(119872)

we have

11987642

leinfin

sum119895=119895lowast+1

sum119896isinΛ 119895

1198892

119895119896le 119862

infin

sum119895=119895lowast+1

2minus2119895119904 le 119862(ln 119899

119899)

2119904(2119904+2120588+1)

(85)


119901119903(119872) and (2119904 + 2120588 + 1)(2 minus


11987642

le 119862

1198951

sum119895=119895lowast+1

1205822minus119901

119895sum

119896isinΛ 119895

10038161003816100381610038161003816119889119895119896

10038161003816100381610038161003816119901

= 119862(ln 119899

119899)

(2minus119901)2 1198951

sum119895=119895lowast+1

2119895120588(2minus119901) sum119896isinΛ 119895

10038161003816100381610038161003816119889119895119896

10038161003816100381610038161003816119901

le 119862(ln 119899

119899)


sum119895=119895lowast+1


le 119862(ln 119899

119899)

(2minus119901)2


le 119862(ln 119899

119899)

2119904(2119904+2120588+1)

(86)


1198764le 119862(

ln 119899

119899)

2119904(2119904+2120588+1)

(87)


119876 le 119862(ln 119899

119899)

2119904(2119904+2120588+1)

(88)





119891 (119909) minus 119891 (119909) = 119878 (119909) minus 119879 (119909) (89)


where

119878 (119909)

=1

119892 (119909)(V (119909) minus V (119909)

+119891 (119909) (119892 (119909) minus 119892 (119909))) 1|119892(119909)|ge119888lowast2

119879 (119909) = 119891 (119909) 1|119892(119909)|lt119888lowast2

(90)


|119878 (119909)| le 119862 (|V (119909) minus V (119909)| + 1003816100381610038161003816119892 (119909) minus 119892 (119909)1003816100381610038161003816) (91)





le 1198621003816100381610038161003816119892 (119909) minus 119892 (119909)

1003816100381610038161003816 (92)



1003816100381610038161003816) (93)


E (10038171003817100381710038171003817119891 minus 119891

100381710038171003817100381710038172

2) le 119862 (E (V minus V2

2) + E (

1003817100381710038171003817119892 minus 11989210038171003817100381710038172

2)) (94)




1are independent

with E(1205851) = 0

E (119902 (120574119895119896

1198811))

= E (1198841120574119895119896

(1198831)) = E (119891 (119883

1) 120574

119895119896(119883

1))

= int1

0

119891 (119909) 120574119895119896

(119909) 119892 (119909) 119889119909 = int1

0

V (119909) 120574119895119896

(119909) 119889119909

(95)

(H2) (i)-(ii)-(iii) with 120588 = 0


|1198841| le 119872 with 119872 gt 0 we have

sup(119909119909lowast)isin1198811(Ω)

10038161003816100381610038161003816119902 (120574119895119896

(119909 119909lowast))

10038161003816100381610038161003816


10038161003816100381610038161003816119909120574119895119896(119909

lowast)10038161003816100381610038161003816


10038161003816100381610038161003816120574119895119896(119909

lowast)10038161003816100381610038161003816 le 11986221198952

(96)


E (10038161003816100381610038161003816119902(120574119895119896

1198811)100381610038161003816100381610038162

) = E (1198842

1(120574

119895119896(119883

1))

2

)

le 119862E ((120574119895119896

(1198831))

2

)

= 119862int1

0

(120574119895119896

(119909))2

119892 (119909) 119889119909

le 119862int1

0

(120574119895119896

(119909))2

119889119909 le 119862

(97)



int1198811(Ω)

10038161003816100381610038161003816119902 (120574119895119896

119909)10038161003816100381610038161003816 119889119909

= (int119872

minus119872

|119909| 119889119909)(int1

0

10038161003816100381610038161003816120574119895119896(119909

lowast)10038161003816100381610038161003816 119889119909lowast

)

= 1198722 int1

0

10038161003816100381610038161003816120574119895119896(119909)

10038161003816100381610038161003816 119889119909 le 1198622minus1198952

(98)



119901119903(119872) with 119903 ge 1 119901 ge 2


E (V minus V22) le 119862(

ln 119899

119899)

2119904(2119904+1)

(99)


119905instead of 119881

119905


119901119903(119872) with 119903 ge 1 119901 ge 2


E (1003817100381710038171003817119892 minus 119892

10038171003817100381710038172

2) le 119862(

ln 119899

119899)

2119904(2119904+1)

(100)




Acknowledgment



References
































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











sup119909isinR

119891 (119909) le 119862 lt infin (18)



sup119909isinR

119906 (119909) le 119862 (19)



1 119881



sup119898isinZ

sup(119909119910)isinR2

119906(1198811 119881119898+1)

(119909 119910) le 119862 (20)


F (120574) (119909) = intinfin

minusinfin

120574 (119910) 119890minus119894119909119910119889119910 119909 isin R (21)



1003816100381610038161003816F (119892) (119909)1003816100381610038161003816 ge

119888

(1 + 1199092)1205752

(22)


100381610038161003816100381610038161003816(F (119892) (119909))

(ℓ)100381610038161003816100381610038161003816le

119862

(1 + |119909|)120575+ℓ (23)



119902 (120574 119909) =1

2120587int

infin

minusinfin

F (120574) (119910)

F (119892) (119910)119890minus119894119910119909119889119910 (24)





E (10038171003817100381710038171003817119891 minus 119891

100381710038171003817100381710038172

2) le 119862(

ln 119899

119899)

2119904(2119904+2120575+1)

(25)








119881119905= 1205902

119905119885

119905 (26)

(1205902




119905and 119885


119905)119905isinZ is a 120572-


1 119881

119899from (119881



119892 (119909) =1

(120575 minus 1)(minus ln119909)

120575minus1 119909 isin [0 1] (27)


119894=1119880

119894 where

1198801 119880





119906 (119909) le 119862 (28)



1 119881



sup(119909119910)isinR2

119906(1198811 119881119898+1)

(119909 119910) le 119862 (29)



119902 (120574 119909) = 119879120575(120574) (119909) (30)


ℓ(120574)(119909) = 119879(119879





E (10038171003817100381710038171003817119891 minus 119891

100381710038171003817100381710038172

2) le 119862(

ln 119899

119899)

2119904(2119904+2120575+1)

(31)





119905 isin Z 119881119905= (119884

119905 119883

119905)

119884119905= 119891 (119883

119905) + 120585

119905 (32)




119905and 120585




1 119881

119899from (119881





L2([0 1])(K2) 120585

1(Ω) is bounded


sup119909isin[01]

1003816100381610038161003816119891 (119909)1003816100381610038161003816 le 119862 (33)


119888lowastle inf

119909isin[01]

119892 (119909) sup119909isin[01]

119892 (119909) le 119862 (34)




119906 (119909) le 119862 (35)



1 119881



sup119898isinZ


119906(1198811 119881119898+1)

(119909 119910) le 119862 (36)



119891 (119909) =V (119909)119892 (119909)

1|119892(119909)|ge119888lowast2

(37)

where


119902 (120574 (119909 119909lowast)) = 119909120574 (119909

lowast) (38)

and 120588 = 0


119905

119902 (120574 119909) = 120574 (119909) (39)

and 120588 = 0(iii) 119888



119901119903(119872) and 119892 isin 119861119904

119901119903(119872) with 119903 ge 1


E (10038171003817100381710038171003817119891 minus 119891

100381710038171003817100381710038172

2) le 119862(

ln 119899

119899)

2119904(2119904+1)

(40)





5 Proofs







1)|119901) and E(|119896(119882

1)|119902) exist


1) 119896 (119882

119898+1))1003816100381610038161003816

le 1198621205721minus1119901minus1119902

119898(E (

1003816100381610038161003816ℎ (1198821)1003816100381610038161003816119901

))1119901

(E(1003816100381610038161003816119896 (119882

1)1003816100381610038161003816119902

))1119902

(41)






119905= ℎ(119882


1) = 0 and



P(1

119899

100381610038161003816100381610038161003816100381610038161003816

119899

sum119894=1

119880119894

100381610038161003816100381610038161003816100381610038161003816ge 120582)

le 4 exp(minus1205822119899

16 (119863119898119898 + 1205821198721198983)

) + 32119872

120582119899120572

119898

(42)

where

119863119898

= max119897isin12119898

V (119897

sum119894=1

119880119894) (43)



10038161003816100381610038161003816C119900V (119902 (120574119895119896

119881119898+1

) 119902 (120574119895119896

1198811))

10038161003816100381610038161003816

=100381610038161003816100381610038161003816100381610038161003816int1198811(Ω)

int1198811(Ω)

119902 (120574119895119896

119909) 119902 (120574119895119896

119910)

times (119906(1198811 119881119898+1)

(119909 119910) minus 119906 (119909) 119906 (119910)) 119889119909119889119910100381610038161003816100381610038161003816100381610038161003816

le int1198811(Ω)

int1198811(Ω)

10038161003816100381610038161003816119902 (120574119895119896

119909)1003816100381610038161003816100381610038161003816100381610038161003816119902 (120574

119895119896 119910)

10038161003816100381610038161003816

times10038161003816100381610038161003816119906(1198811 119881119898+1)

(119909 119910) minus 119906 (119909) 119906 (119910)10038161003816100381610038161003816 119889119909 119889119910

le 119862(int1198811(Ω)

10038161003816100381610038161003816119902 (120574119895119896

119909)10038161003816100381610038161003816 119889119909)

2

le 119862221205881198952minus119895

(44)




and 120573119895119896


0be (9) and let 119895

1be (10)


0 119895


119895

E (10038161003816100381610038161003816119888119895119896 minus 119888

119895119896

100381610038161003816100381610038162

) le 11986222120588119895 1

119899 (45)

E (10038161003816100381610038161003816119889119895119896

minus 119889119895119896

100381610038161003816100381610038162

) le 11986222120588119895 1

119899 (46)


0 119895


119895

E (10038161003816100381610038161003816119889119895119896

minus 119889119895119896

100381610038161003816100381610038164

) le 11986224120588119895 (47)



1 and


P(10038161003816100381610038161003816119889119895119896

minus 119889119895119896

10038161003816100381610038161003816 ge120581120582

119895

2) le 119862

1

1198994 (48)



E (10038161003816100381610038161003816119888119895119896 minus 119888

119895119896

100381610038161003816100381610038162

) = V (119888119895119896

)

=1

1198992

119899

sum119894=1

V (119902 (120601119895119896

119881119894)) +

2

1198992

times119899

sumV=2

Vminus1

sumℓ=1

Re (C119900V (119902 (120601

119895119896 119881V) 119902 (120601

119895119896 119881

ℓ)))

=1

1198992

119899

sum119894=1

V (119902 (120601119895k 119881119894

)) +2

1198992

times119899minus1

sum119898=1

(119899 minus 119898)Re (C119900V (119902 (120601

119895119896 119881

119898+1) 119902 (120601

119895119896 119881

1)))

le1

119899(E (

10038161003816100381610038161003816119902 (120601119895119896

1198811)100381610038161003816100381610038162

)

+2119899minus1

sum119898=1

10038161003816100381610038161003816C119900V (119902 (120601119895119896

119881119898+1

) 119902 (120601119895119896

1198811))

10038161003816100381610038161003816)

(49)

By (H2)-(ii) we get

E (10038161003816100381610038161003816119902(120601119895119896

1198811)100381610038161003816100381610038162

) le 11986222120588119895 (50)


119899minus1

sum119898=1

10038161003816100381610038161003816C119900V (119902 (120601119895119896

119881119898+1

) 119902 (120601119895119896

1198811))

10038161003816100381610038161003816 = 119860 + 119861 (51)

where

119860 =[ln 119899120579]minus1

sum119898=1

10038161003816100381610038161003816C119900V (119902 (120601119895119896

119881119898+1

) 119902 (120601119895119896

1198811))

10038161003816100381610038161003816

119861 =119899minus1

sum119898=[(ln 119899)120579]

10038161003816100381610038161003816C119900V (119902 (120601119895119896

119881119898+1

) 119902 (120601119895119896

1198811))

10038161003816100381610038161003816

(52)


119860 le 119862221205881198952minus119895 [ln 119899

120579] le 11986222120588119895 (53)




119861 le 119862radicE (10038161003816100381610038161003816119902(120601119895119896

1198811)100381610038161003816100381610038164

)119899minus1

sum119898=[(ln 119899)120579]

radic120572119898

le 119862212058811989521198952radicE (10038161003816100381610038161003816119902(120601119895119896

1198811)100381610038161003816100381610038162

)infin

sum119898=[(ln 119899)120579]

119890minus1205791198982

= 11986222120588119895radic119899119890minus(ln 119899)2 le 11986222120588119895

(54)

Thus119899minus1

sum119898=1

10038161003816100381610038161003816C119900V (119902 (120601119895119896

119881119898+1

) 119902 (120601119895119896

1198811))

10038161003816100381610038161003816 le 11986222120588119895 (55)



| le

sup119909isin1198811(Ω)

|119902(120595119895119896


119895119896| le 119891

2le 119862 that

10038161003816100381610038161003816119889119895119896minus 119889

119895119896

10038161003816100381610038161003816 le10038161003816100381610038161003816119889119895119896

10038161003816100381610038161003816 +10038161003816100381610038161003816119889119895119896

10038161003816100381610038161003816 le 119862212058811989521198952 (56)


E (10038161003816100381610038161003816119889119895119896

minus 119889119895119896

100381610038161003816100381610038164

) le 119862221205881198952119895E (

10038161003816100381610038161003816119889119895119896minus 119889

119895119896

100381610038161003816100381610038162

) le 119862241205881198952119895 1

119899

(57)


Lemma 9 Let us set

119880119894= 119902 (120595

119895119896 119881

119894) minus E (119902 (120595

119895119896 119881

1)) (58)


119899(ln 119899)3

1003816100381610038161003816119880119894

1003816100381610038161003816 le 2 sup119909isin1198811(Ω)

10038161003816100381610038161003816119902 (120595119895119896

119909)10038161003816100381610038161003816 le 119862212058811989521198952 le 1198622120588119895

radic119899

(ln 119899)3

(59)



V (119897

sum119894=1

119880119894)

= V (119897

sum119894=1

119902 (120595119895119896

119881119894)) le 11986222120588119895 (119897 + 11989722minus119895) le 11986222120588119895119897

(60)

Therefore

119863119898

= max119897isin12119898

V (119897

sum119894=1

119880119894) le 11986222120588119895119898 (61)


119899 120582 = 120581120582

1198952


P(10038161003816100381610038161003816119889119895119896

minus 119889119895119896

10038161003816100381610038161003816 ge120581120582

119895

2)

le 119862(exp(minus11986212058121205822

119895119899

119863119898119898 + 120581120582

119895119898119872

) +119872

120582119895

119899119890minus120579119898)

le 119862( exp (minus119862((120581222120588119895 ln 119899)

times (22120588119895 + 1205812120588119895radic (ln 119899)

119899

times [radic120581 ln 119899] 2120588119895

radic119899

(ln 119899)3)

minus1

))

+ radic119899(ln 119899)3

(ln 119899) 119899119899119890minus120579[radic120581 ln 119899])

le 119862(119899minus1198621205812(1+120581

32) + 1198992minus120579radic120581)

(62)





119891 (119909) = sum119896isinΛ 1198950

1198881198950 119896

1206011198950 119896

(119909) +infin

sum119895=1198950

sum119896isinΛ 119895

119889119895119896

120595119895119896

(119909) (63)

where 1198881198950 119896

= int1

0119891(119909)120601

1198950 119896(119909)119889119909 and 119889

119895119896= int

1

0119891(119909)120595

119895119896(119909)119889119909


E (10038171003817100381710038171003817119891 minus 119891

100381710038171003817100381710038172

2) = 119875 + 119876 + 119877 (64)

where

119875 = sum119896isinΛ 1198950

E (100381610038161003816100381610038161198881198950 119896 minus 119888

1198950 119896

100381610038161003816100381610038162

)

119876 =

1198951

sum119895=1198950

sum119896isinΛ 119895

E(100381610038161003816100381610038161003816119889

1198951198961|119889119895119896|ge120581120582119895

minus 119889119895119896

100381610038161003816100381610038161003816

2

)

119877 =infin

sum119895=1198951+1

sum119896isinΛ 119895

1198892

119895119896

(65)




119875 le 11986221198950

119899le 119862

ln 119899

119899le 119862(

ln 119899

119899)

2119904(2119904+2120588+1)

(66)


119901119903(119872) sube 119861119904

2infin(119872)


119877 le 119862infin

sum119895=1198951+1

2minus2119895119904 le 119862((ln 119899)3

119899)

2119904(2120588+1)

le 119862(ln 119899

119899)

2119904(2119904+2120588+1)

(67)


119901119903(119872) sube

119861119904+12minus1119901



119877 le 119862infin

sum119895=1198951+1

2minus2119895(119904+12minus1119901)

le 119862((ln 119899)3

119899)

2(119904+12minus1119901)(2120588+1)

le 119862(ln 119899

119899)

2119904(2119904+2120588+1)

(68)


119877 le 119862(ln 119899

119899)

2119904(2119904+2120588+1)

(69)


= 119889119895119896

minus 119889119895119896


119876 =4

sum119894=1

119876119894 (70)

where

1198761=

1198951

sum119895=1198950

sum119896isinΛ 119895

E (10038161003816100381610038161003816119863119895119896

100381610038161003816100381610038162

1|119889119895119896|ge120581120582119895 |119889119895119896|lt1205811205821198952

)

1198762=

1198951

sum119895=1198950

sum119896isinΛ 119895

E (10038161003816100381610038161003816119863119895119896

100381610038161003816100381610038162

1|119889119895119896|ge120581120582119895 |119889119895119896|ge1205811205821198952

)

1198763=

1198951

sum119895=1198950

sum119896isinΛ 119895

E (1198892

1198951198961|119889119895119896|lt120581120582119895 |119889119895119896|ge2120581120582119895

)

1198764=

1198951

sum119895=1198950

sum119896isinΛ 119895

E (1198892

1198951198961|119889119895119896|lt120581120582119895 |119889119895119896|lt2120581120582119895

)

(71)



1|119889119895119896|lt120581120582119895 |119889119895119896|ge2120581120582119895

le 1|119895119896|gt1205811205821198952

1|119889119895119896|ge120581120582119895 |119889119895119896|lt1205811205821198952

le

1|119895119896|gt1205811205821198952

and 1|119889119895119896|lt120581120582119895 |119889119895119896|ge2120581120582119895

le 1|119889119895119896|le2|119895119896|


1198761+ 119876

3le 119862

1198951

sum119895=1198950

sum119896isinΛ 119895

E (10038161003816100381610038161003816119863119895119896

100381610038161003816100381610038162

1|119895119896|gt1205811205821198952

)

le 119862

1198951

sum119895=1198950

sum119896isinΛ 119895

(E (10038161003816100381610038161003816119863119895119896

100381610038161003816100381610038164

))12

(P(10038161003816100381610038161003816119863119895119896

10038161003816100381610038161003816 gt120581120582

119895

2))

12

le 1198621

1198992

1198951

sum119895=1198950

2119895(1+2120588) le 1198621

119899le 119862(

ln 119899

119899)

2119904(2119904+2120588+1)

(72)


Proposition 10 that

1198762le

1198951

sum119895=1198950

sum119896isinΛ 119895

E (10038161003816100381610038161003816119863119895119896

100381610038161003816100381610038162

) 1|119889119895119896|ge1205811205821198952

le 1198621

119899

1198951

sum119895=j0

22120588119895 sum119896isinΛ 119895

1|119889119895119896|gt1205811205821198952

(73)


2119895lowast = [(119899

ln 119899)

1(2119904+2120588+1)

] (74)


0 119895



1198762le 119876

21+ 119876

22 (75)

where

11987621

= 1198621

119899

119895lowast

sum119895=1198950

22120588119895 sum119896isinΛ 119895

1|119889119895119896|gt1205811205821198952

11987622

= 1198621

119899

1198951

sum119895=119895lowast+1

22120588119895 sum119896isinΛ 119895

1|119889119895119896|gt1205811205821198952

(76)


11987621

le 119862ln 119899

119899

119895lowast

sum119895=1198950

2119895(1+2120588) le 119862(ln 119899

119899)

2119904(2119904+2120588+1)

(77)




119901119903(119872) sube 119861119904


11987622

le 119862ln 119899

119899

1198951

sum119895=119895lowast+1

22120588119895 1

1205822

119895

sum119896isinΛ 119895

1198892

119895119896le 119862

infin

sum119895=119895lowast+1

sum119896isinΛ 119895

1198892

119895119896

le 119862infin

sum119895=119895lowast+1

2minus2119895119904 le 119862(ln 119899

119899)

2119904(2119904+2120588+1)

(78)


119901119903(119872) and (2119904 + 2120588 + 1)(2 minus 119901)2 +


11987622

le 119862ln 119899

119899

1198951

sum119895=119895lowast+1

22120588119895 1

120582119901

119895

sum119896isinΛ 119895

10038161003816100381610038161003816119889119895119896

10038161003816100381610038161003816119901

le 119862(ln 119899

119899)


sum119895=119895lowast+1


le 119862(ln 119899

119899)

(2minus119901)2


le 119862(ln 119899

119899)

2119904(2119904+2120588+1)

(79)


1198762le C(

ln 119899

119899)

2119904(2119904+2120588+1)

(80)


1198764le

1198951

sum119895=1198950

sum119896isinΛ 119895

1198892

1198951198961|119889119895119896|lt2120581120582119895

(81)


4can be bound as

1198764le 119876

41+ 119876

42 (82)

where

11987641

=

119895lowast

sum119895=1198950

sum119896isinΛ 119895

1198892

1198951198961|119889119895119896|lt2120581120582119895

11987642

=

1198951

sum119895=119895lowast+1

sum119896isinΛ 119895

1198892

1198951198961|119889119895119896|lt2120581120582119895

(83)


11987641

le 119862

119895lowast

sum119895=1198950

21198951205822

119895= 119862

ln 119899

119899

119895lowast

sum119895=1198950

2119895(1+2120588) le 119862(ln 119899

119899)

2119904(2119904+2120588+1)

(84)



119901119903(119872) sube 119861119904

2infin(119872)

we have

11987642

leinfin

sum119895=119895lowast+1

sum119896isinΛ 119895

1198892

119895119896le 119862

infin

sum119895=119895lowast+1

2minus2119895119904 le 119862(ln 119899

119899)

2119904(2119904+2120588+1)

(85)


119901119903(119872) and (2119904 + 2120588 + 1)(2 minus


11987642

le 119862

1198951

sum119895=119895lowast+1

1205822minus119901

119895sum

119896isinΛ 119895

10038161003816100381610038161003816119889119895119896

10038161003816100381610038161003816119901

= 119862(ln 119899

119899)

(2minus119901)2 1198951

sum119895=119895lowast+1

2119895120588(2minus119901) sum119896isinΛ 119895

10038161003816100381610038161003816119889119895119896

10038161003816100381610038161003816119901

le 119862(ln 119899

119899)


sum119895=119895lowast+1


le 119862(ln 119899

119899)

(2minus119901)2


le 119862(ln 119899

119899)

2119904(2119904+2120588+1)

(86)


1198764le 119862(

ln 119899

119899)

2119904(2119904+2120588+1)

(87)


119876 le 119862(ln 119899

119899)

2119904(2119904+2120588+1)

(88)





119891 (119909) minus 119891 (119909) = 119878 (119909) minus 119879 (119909) (89)


where

119878 (119909)

=1

119892 (119909)(V (119909) minus V (119909)

+119891 (119909) (119892 (119909) minus 119892 (119909))) 1|119892(119909)|ge119888lowast2

119879 (119909) = 119891 (119909) 1|119892(119909)|lt119888lowast2

(90)


|119878 (119909)| le 119862 (|V (119909) minus V (119909)| + 1003816100381610038161003816119892 (119909) minus 119892 (119909)1003816100381610038161003816) (91)





le 1198621003816100381610038161003816119892 (119909) minus 119892 (119909)

1003816100381610038161003816 (92)



1003816100381610038161003816) (93)


E (10038171003817100381710038171003817119891 minus 119891

100381710038171003817100381710038172

2) le 119862 (E (V minus V2

2) + E (

1003817100381710038171003817119892 minus 11989210038171003817100381710038172

2)) (94)




1are independent

with E(1205851) = 0

E (119902 (120574119895119896

1198811))

= E (1198841120574119895119896

(1198831)) = E (119891 (119883

1) 120574

119895119896(119883

1))

= int1

0

119891 (119909) 120574119895119896

(119909) 119892 (119909) 119889119909 = int1

0

V (119909) 120574119895119896

(119909) 119889119909

(95)

(H2) (i)-(ii)-(iii) with 120588 = 0


|1198841| le 119872 with 119872 gt 0 we have

sup(119909119909lowast)isin1198811(Ω)

10038161003816100381610038161003816119902 (120574119895119896

(119909 119909lowast))

10038161003816100381610038161003816


10038161003816100381610038161003816119909120574119895119896(119909

lowast)10038161003816100381610038161003816


10038161003816100381610038161003816120574119895119896(119909

lowast)10038161003816100381610038161003816 le 11986221198952

(96)


E (10038161003816100381610038161003816119902(120574119895119896

1198811)100381610038161003816100381610038162

) = E (1198842

1(120574

119895119896(119883

1))

2

)

le 119862E ((120574119895119896

(1198831))

2

)

= 119862int1

0

(120574119895119896

(119909))2

119892 (119909) 119889119909

le 119862int1

0

(120574119895119896

(119909))2

119889119909 le 119862

(97)



int1198811(Ω)

10038161003816100381610038161003816119902 (120574119895119896

119909)10038161003816100381610038161003816 119889119909

= (int119872

minus119872

|119909| 119889119909)(int1

0

10038161003816100381610038161003816120574119895119896(119909

lowast)10038161003816100381610038161003816 119889119909lowast

)

= 1198722 int1

0

10038161003816100381610038161003816120574119895119896(119909)

10038161003816100381610038161003816 119889119909 le 1198622minus1198952

(98)



119901119903(119872) with 119903 ge 1 119901 ge 2


E (V minus V22) le 119862(

ln 119899

119899)

2119904(2119904+1)

(99)


119905instead of 119881

119905


119901119903(119872) with 119903 ge 1 119901 ge 2


E (1003817100381710038171003817119892 minus 119892

10038171003817100381710038172

2) le 119862(

ln 119899

119899)

2119904(2119904+1)

(100)




Acknowledgment



References
































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119905 isin Z 119881119905= (119884

119905 119883

119905)

119884119905= 119891 (119883

119905) + 120585

119905 (32)




119905and 120585




1 119881

119899from (119881





L2([0 1])(K2) 120585

1(Ω) is bounded


sup119909isin[01]

1003816100381610038161003816119891 (119909)1003816100381610038161003816 le 119862 (33)


119888lowastle inf

119909isin[01]

119892 (119909) sup119909isin[01]

119892 (119909) le 119862 (34)




119906 (119909) le 119862 (35)



1 119881



sup119898isinZ


119906(1198811 119881119898+1)

(119909 119910) le 119862 (36)



119891 (119909) =V (119909)119892 (119909)

1|119892(119909)|ge119888lowast2

(37)

where


119902 (120574 (119909 119909lowast)) = 119909120574 (119909

lowast) (38)

and 120588 = 0


119905

119902 (120574 119909) = 120574 (119909) (39)

and 120588 = 0(iii) 119888



119901119903(119872) and 119892 isin 119861119904

119901119903(119872) with 119903 ge 1


E (10038171003817100381710038171003817119891 minus 119891

100381710038171003817100381710038172

2) le 119862(

ln 119899

119899)

2119904(2119904+1)

(40)





5 Proofs







1)|119901) and E(|119896(119882

1)|119902) exist


1) 119896 (119882

119898+1))1003816100381610038161003816

le 1198621205721minus1119901minus1119902

119898(E (

1003816100381610038161003816ℎ (1198821)1003816100381610038161003816119901

))1119901

(E(1003816100381610038161003816119896 (119882

1)1003816100381610038161003816119902

))1119902

(41)






119905= ℎ(119882


1) = 0 and



P(1

119899

100381610038161003816100381610038161003816100381610038161003816

119899

sum119894=1

119880119894

100381610038161003816100381610038161003816100381610038161003816ge 120582)

le 4 exp(minus1205822119899

16 (119863119898119898 + 1205821198721198983)

) + 32119872

120582119899120572

119898

(42)

where

119863119898

= max119897isin12119898

V (119897

sum119894=1

119880119894) (43)



10038161003816100381610038161003816C119900V (119902 (120574119895119896

119881119898+1

) 119902 (120574119895119896

1198811))

10038161003816100381610038161003816

=100381610038161003816100381610038161003816100381610038161003816int1198811(Ω)

int1198811(Ω)

119902 (120574119895119896

119909) 119902 (120574119895119896

119910)

times (119906(1198811 119881119898+1)

(119909 119910) minus 119906 (119909) 119906 (119910)) 119889119909119889119910100381610038161003816100381610038161003816100381610038161003816

le int1198811(Ω)

int1198811(Ω)

10038161003816100381610038161003816119902 (120574119895119896

119909)1003816100381610038161003816100381610038161003816100381610038161003816119902 (120574

119895119896 119910)

10038161003816100381610038161003816

times10038161003816100381610038161003816119906(1198811 119881119898+1)

(119909 119910) minus 119906 (119909) 119906 (119910)10038161003816100381610038161003816 119889119909 119889119910

le 119862(int1198811(Ω)

10038161003816100381610038161003816119902 (120574119895119896

119909)10038161003816100381610038161003816 119889119909)

2

le 119862221205881198952minus119895

(44)




and 120573119895119896


0be (9) and let 119895

1be (10)


0 119895


119895

E (10038161003816100381610038161003816119888119895119896 minus 119888

119895119896

100381610038161003816100381610038162

) le 11986222120588119895 1

119899 (45)

E (10038161003816100381610038161003816119889119895119896

minus 119889119895119896

100381610038161003816100381610038162

) le 11986222120588119895 1

119899 (46)


0 119895


119895

E (10038161003816100381610038161003816119889119895119896

minus 119889119895119896

100381610038161003816100381610038164

) le 11986224120588119895 (47)



1 and


P(10038161003816100381610038161003816119889119895119896

minus 119889119895119896

10038161003816100381610038161003816 ge120581120582

119895

2) le 119862

1

1198994 (48)



E (10038161003816100381610038161003816119888119895119896 minus 119888

119895119896

100381610038161003816100381610038162

) = V (119888119895119896

)

=1

1198992

119899

sum119894=1

V (119902 (120601119895119896

119881119894)) +

2

1198992

times119899

sumV=2

Vminus1

sumℓ=1

Re (C119900V (119902 (120601

119895119896 119881V) 119902 (120601

119895119896 119881

ℓ)))

=1

1198992

119899

sum119894=1

V (119902 (120601119895k 119881119894

)) +2

1198992

times119899minus1

sum119898=1

(119899 minus 119898)Re (C119900V (119902 (120601

119895119896 119881

119898+1) 119902 (120601

119895119896 119881

1)))

le1

119899(E (

10038161003816100381610038161003816119902 (120601119895119896

1198811)100381610038161003816100381610038162

)

+2119899minus1

sum119898=1

10038161003816100381610038161003816C119900V (119902 (120601119895119896

119881119898+1

) 119902 (120601119895119896

1198811))

10038161003816100381610038161003816)

(49)

By (H2)-(ii) we get

E (10038161003816100381610038161003816119902(120601119895119896

1198811)100381610038161003816100381610038162

) le 11986222120588119895 (50)


119899minus1

sum119898=1

10038161003816100381610038161003816C119900V (119902 (120601119895119896

119881119898+1

) 119902 (120601119895119896

1198811))

10038161003816100381610038161003816 = 119860 + 119861 (51)

where

119860 =[ln 119899120579]minus1

sum119898=1

10038161003816100381610038161003816C119900V (119902 (120601119895119896

119881119898+1

) 119902 (120601119895119896

1198811))

10038161003816100381610038161003816

119861 =119899minus1

sum119898=[(ln 119899)120579]

10038161003816100381610038161003816C119900V (119902 (120601119895119896

119881119898+1

) 119902 (120601119895119896

1198811))

10038161003816100381610038161003816

(52)


119860 le 119862221205881198952minus119895 [ln 119899

120579] le 11986222120588119895 (53)




119861 le 119862radicE (10038161003816100381610038161003816119902(120601119895119896

1198811)100381610038161003816100381610038164

)119899minus1

sum119898=[(ln 119899)120579]

radic120572119898

le 119862212058811989521198952radicE (10038161003816100381610038161003816119902(120601119895119896

1198811)100381610038161003816100381610038162

)infin

sum119898=[(ln 119899)120579]

119890minus1205791198982

= 11986222120588119895radic119899119890minus(ln 119899)2 le 11986222120588119895

(54)

Thus119899minus1

sum119898=1

10038161003816100381610038161003816C119900V (119902 (120601119895119896

119881119898+1

) 119902 (120601119895119896

1198811))

10038161003816100381610038161003816 le 11986222120588119895 (55)



| le

sup119909isin1198811(Ω)

|119902(120595119895119896


119895119896| le 119891

2le 119862 that

10038161003816100381610038161003816119889119895119896minus 119889

119895119896

10038161003816100381610038161003816 le10038161003816100381610038161003816119889119895119896

10038161003816100381610038161003816 +10038161003816100381610038161003816119889119895119896

10038161003816100381610038161003816 le 119862212058811989521198952 (56)


E (10038161003816100381610038161003816119889119895119896

minus 119889119895119896

100381610038161003816100381610038164

) le 119862221205881198952119895E (

10038161003816100381610038161003816119889119895119896minus 119889

119895119896

100381610038161003816100381610038162

) le 119862241205881198952119895 1

119899

(57)


Lemma 9 Let us set

119880119894= 119902 (120595

119895119896 119881

119894) minus E (119902 (120595

119895119896 119881

1)) (58)


119899(ln 119899)3

1003816100381610038161003816119880119894

1003816100381610038161003816 le 2 sup119909isin1198811(Ω)

10038161003816100381610038161003816119902 (120595119895119896

119909)10038161003816100381610038161003816 le 119862212058811989521198952 le 1198622120588119895

radic119899

(ln 119899)3

(59)



V (119897

sum119894=1

119880119894)

= V (119897

sum119894=1

119902 (120595119895119896

119881119894)) le 11986222120588119895 (119897 + 11989722minus119895) le 11986222120588119895119897

(60)

Therefore

119863119898

= max119897isin12119898

V (119897

sum119894=1

119880119894) le 11986222120588119895119898 (61)


119899 120582 = 120581120582

1198952


P(10038161003816100381610038161003816119889119895119896

minus 119889119895119896

10038161003816100381610038161003816 ge120581120582

119895

2)

le 119862(exp(minus11986212058121205822

119895119899

119863119898119898 + 120581120582

119895119898119872

) +119872

120582119895

119899119890minus120579119898)

le 119862( exp (minus119862((120581222120588119895 ln 119899)

times (22120588119895 + 1205812120588119895radic (ln 119899)

119899

times [radic120581 ln 119899] 2120588119895

radic119899

(ln 119899)3)

minus1

))

+ radic119899(ln 119899)3

(ln 119899) 119899119899119890minus120579[radic120581 ln 119899])

le 119862(119899minus1198621205812(1+120581

32) + 1198992minus120579radic120581)

(62)





119891 (119909) = sum119896isinΛ 1198950

1198881198950 119896

1206011198950 119896

(119909) +infin

sum119895=1198950

sum119896isinΛ 119895

119889119895119896

120595119895119896

(119909) (63)

where 1198881198950 119896

= int1

0119891(119909)120601

1198950 119896(119909)119889119909 and 119889

119895119896= int

1

0119891(119909)120595

119895119896(119909)119889119909


E (10038171003817100381710038171003817119891 minus 119891

100381710038171003817100381710038172

2) = 119875 + 119876 + 119877 (64)

where

119875 = sum119896isinΛ 1198950

E (100381610038161003816100381610038161198881198950 119896 minus 119888

1198950 119896

100381610038161003816100381610038162

)

119876 =

1198951

sum119895=1198950

sum119896isinΛ 119895

E(100381610038161003816100381610038161003816119889

1198951198961|119889119895119896|ge120581120582119895

minus 119889119895119896

100381610038161003816100381610038161003816

2

)

119877 =infin

sum119895=1198951+1

sum119896isinΛ 119895

1198892

119895119896

(65)




119875 le 11986221198950

119899le 119862

ln 119899

119899le 119862(

ln 119899

119899)

2119904(2119904+2120588+1)

(66)


119901119903(119872) sube 119861119904

2infin(119872)


119877 le 119862infin

sum119895=1198951+1

2minus2119895119904 le 119862((ln 119899)3

119899)

2119904(2120588+1)

le 119862(ln 119899

119899)

2119904(2119904+2120588+1)

(67)


119901119903(119872) sube

119861119904+12minus1119901



119877 le 119862infin

sum119895=1198951+1

2minus2119895(119904+12minus1119901)

le 119862((ln 119899)3

119899)

2(119904+12minus1119901)(2120588+1)

le 119862(ln 119899

119899)

2119904(2119904+2120588+1)

(68)


119877 le 119862(ln 119899

119899)

2119904(2119904+2120588+1)

(69)


= 119889119895119896

minus 119889119895119896


119876 =4

sum119894=1

119876119894 (70)

where

1198761=

1198951

sum119895=1198950

sum119896isinΛ 119895

E (10038161003816100381610038161003816119863119895119896

100381610038161003816100381610038162

1|119889119895119896|ge120581120582119895 |119889119895119896|lt1205811205821198952

)

1198762=

1198951

sum119895=1198950

sum119896isinΛ 119895

E (10038161003816100381610038161003816119863119895119896

100381610038161003816100381610038162

1|119889119895119896|ge120581120582119895 |119889119895119896|ge1205811205821198952

)

1198763=

1198951

sum119895=1198950

sum119896isinΛ 119895

E (1198892

1198951198961|119889119895119896|lt120581120582119895 |119889119895119896|ge2120581120582119895

)

1198764=

1198951

sum119895=1198950

sum119896isinΛ 119895

E (1198892

1198951198961|119889119895119896|lt120581120582119895 |119889119895119896|lt2120581120582119895

)

(71)



1|119889119895119896|lt120581120582119895 |119889119895119896|ge2120581120582119895

le 1|119895119896|gt1205811205821198952

1|119889119895119896|ge120581120582119895 |119889119895119896|lt1205811205821198952

le

1|119895119896|gt1205811205821198952

and 1|119889119895119896|lt120581120582119895 |119889119895119896|ge2120581120582119895

le 1|119889119895119896|le2|119895119896|


1198761+ 119876

3le 119862

1198951

sum119895=1198950

sum119896isinΛ 119895

E (10038161003816100381610038161003816119863119895119896

100381610038161003816100381610038162

1|119895119896|gt1205811205821198952

)

le 119862

1198951

sum119895=1198950

sum119896isinΛ 119895

(E (10038161003816100381610038161003816119863119895119896

100381610038161003816100381610038164

))12

(P(10038161003816100381610038161003816119863119895119896

10038161003816100381610038161003816 gt120581120582

119895

2))

12

le 1198621

1198992

1198951

sum119895=1198950

2119895(1+2120588) le 1198621

119899le 119862(

ln 119899

119899)

2119904(2119904+2120588+1)

(72)


Proposition 10 that

1198762le

1198951

sum119895=1198950

sum119896isinΛ 119895

E (10038161003816100381610038161003816119863119895119896

100381610038161003816100381610038162

) 1|119889119895119896|ge1205811205821198952

le 1198621

119899

1198951

sum119895=j0

22120588119895 sum119896isinΛ 119895

1|119889119895119896|gt1205811205821198952

(73)


2119895lowast = [(119899

ln 119899)

1(2119904+2120588+1)

] (74)


0 119895



1198762le 119876

21+ 119876

22 (75)

where

11987621

= 1198621

119899

119895lowast

sum119895=1198950

22120588119895 sum119896isinΛ 119895

1|119889119895119896|gt1205811205821198952

11987622

= 1198621

119899

1198951

sum119895=119895lowast+1

22120588119895 sum119896isinΛ 119895

1|119889119895119896|gt1205811205821198952

(76)


11987621

le 119862ln 119899

119899

119895lowast

sum119895=1198950

2119895(1+2120588) le 119862(ln 119899

119899)

2119904(2119904+2120588+1)

(77)




119901119903(119872) sube 119861119904


11987622

le 119862ln 119899

119899

1198951

sum119895=119895lowast+1

22120588119895 1

1205822

119895

sum119896isinΛ 119895

1198892

119895119896le 119862

infin

sum119895=119895lowast+1

sum119896isinΛ 119895

1198892

119895119896

le 119862infin

sum119895=119895lowast+1

2minus2119895119904 le 119862(ln 119899

119899)

2119904(2119904+2120588+1)

(78)


119901119903(119872) and (2119904 + 2120588 + 1)(2 minus 119901)2 +


11987622

le 119862ln 119899

119899

1198951

sum119895=119895lowast+1

22120588119895 1

120582119901

119895

sum119896isinΛ 119895

10038161003816100381610038161003816119889119895119896

10038161003816100381610038161003816119901

le 119862(ln 119899

119899)


sum119895=119895lowast+1


le 119862(ln 119899

119899)

(2minus119901)2


le 119862(ln 119899

119899)

2119904(2119904+2120588+1)

(79)


1198762le C(

ln 119899

119899)

2119904(2119904+2120588+1)

(80)


1198764le

1198951

sum119895=1198950

sum119896isinΛ 119895

1198892

1198951198961|119889119895119896|lt2120581120582119895

(81)


4can be bound as

1198764le 119876

41+ 119876

42 (82)

where

11987641

=

119895lowast

sum119895=1198950

sum119896isinΛ 119895

1198892

1198951198961|119889119895119896|lt2120581120582119895

11987642

=

1198951

sum119895=119895lowast+1

sum119896isinΛ 119895

1198892

1198951198961|119889119895119896|lt2120581120582119895

(83)


11987641

le 119862

119895lowast

sum119895=1198950

21198951205822

119895= 119862

ln 119899

119899

119895lowast

sum119895=1198950

2119895(1+2120588) le 119862(ln 119899

119899)

2119904(2119904+2120588+1)

(84)



119901119903(119872) sube 119861119904

2infin(119872)

we have

11987642

leinfin

sum119895=119895lowast+1

sum119896isinΛ 119895

1198892

119895119896le 119862

infin

sum119895=119895lowast+1

2minus2119895119904 le 119862(ln 119899

119899)

2119904(2119904+2120588+1)

(85)


119901119903(119872) and (2119904 + 2120588 + 1)(2 minus


11987642

le 119862

1198951

sum119895=119895lowast+1

1205822minus119901

119895sum

119896isinΛ 119895

10038161003816100381610038161003816119889119895119896

10038161003816100381610038161003816119901

= 119862(ln 119899

119899)

(2minus119901)2 1198951

sum119895=119895lowast+1

2119895120588(2minus119901) sum119896isinΛ 119895

10038161003816100381610038161003816119889119895119896

10038161003816100381610038161003816119901

le 119862(ln 119899

119899)


sum119895=119895lowast+1


le 119862(ln 119899

119899)

(2minus119901)2


le 119862(ln 119899

119899)

2119904(2119904+2120588+1)

(86)


1198764le 119862(

ln 119899

119899)

2119904(2119904+2120588+1)

(87)


119876 le 119862(ln 119899

119899)

2119904(2119904+2120588+1)

(88)





119891 (119909) minus 119891 (119909) = 119878 (119909) minus 119879 (119909) (89)


where

119878 (119909)

=1

119892 (119909)(V (119909) minus V (119909)

+119891 (119909) (119892 (119909) minus 119892 (119909))) 1|119892(119909)|ge119888lowast2

119879 (119909) = 119891 (119909) 1|119892(119909)|lt119888lowast2

(90)


|119878 (119909)| le 119862 (|V (119909) minus V (119909)| + 1003816100381610038161003816119892 (119909) minus 119892 (119909)1003816100381610038161003816) (91)





le 1198621003816100381610038161003816119892 (119909) minus 119892 (119909)

1003816100381610038161003816 (92)



1003816100381610038161003816) (93)


E (10038171003817100381710038171003817119891 minus 119891

100381710038171003817100381710038172

2) le 119862 (E (V minus V2

2) + E (

1003817100381710038171003817119892 minus 11989210038171003817100381710038172

2)) (94)




1are independent

with E(1205851) = 0

E (119902 (120574119895119896

1198811))

= E (1198841120574119895119896

(1198831)) = E (119891 (119883

1) 120574

119895119896(119883

1))

= int1

0

119891 (119909) 120574119895119896

(119909) 119892 (119909) 119889119909 = int1

0

V (119909) 120574119895119896

(119909) 119889119909

(95)

(H2) (i)-(ii)-(iii) with 120588 = 0


|1198841| le 119872 with 119872 gt 0 we have

sup(119909119909lowast)isin1198811(Ω)

10038161003816100381610038161003816119902 (120574119895119896

(119909 119909lowast))

10038161003816100381610038161003816


10038161003816100381610038161003816119909120574119895119896(119909

lowast)10038161003816100381610038161003816


10038161003816100381610038161003816120574119895119896(119909

lowast)10038161003816100381610038161003816 le 11986221198952

(96)


E (10038161003816100381610038161003816119902(120574119895119896

1198811)100381610038161003816100381610038162

) = E (1198842

1(120574

119895119896(119883

1))

2

)

le 119862E ((120574119895119896

(1198831))

2

)

= 119862int1

0

(120574119895119896

(119909))2

119892 (119909) 119889119909

le 119862int1

0

(120574119895119896

(119909))2

119889119909 le 119862

(97)



int1198811(Ω)

10038161003816100381610038161003816119902 (120574119895119896

119909)10038161003816100381610038161003816 119889119909

= (int119872

minus119872

|119909| 119889119909)(int1

0

10038161003816100381610038161003816120574119895119896(119909

lowast)10038161003816100381610038161003816 119889119909lowast

)

= 1198722 int1

0

10038161003816100381610038161003816120574119895119896(119909)

10038161003816100381610038161003816 119889119909 le 1198622minus1198952

(98)



119901119903(119872) with 119903 ge 1 119901 ge 2


E (V minus V22) le 119862(

ln 119899

119899)

2119904(2119904+1)

(99)


119905instead of 119881

119905


119901119903(119872) with 119903 ge 1 119901 ge 2


E (1003817100381710038171003817119892 minus 119892

10038171003817100381710038172

2) le 119862(

ln 119899

119899)

2119904(2119904+1)

(100)




Acknowledgment



References
































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













119905= ℎ(119882


1) = 0 and



P(1

119899

100381610038161003816100381610038161003816100381610038161003816

119899

sum119894=1

119880119894

100381610038161003816100381610038161003816100381610038161003816ge 120582)

le 4 exp(minus1205822119899

16 (119863119898119898 + 1205821198721198983)

) + 32119872

120582119899120572

119898

(42)

where

119863119898

= max119897isin12119898

V (119897

sum119894=1

119880119894) (43)



10038161003816100381610038161003816C119900V (119902 (120574119895119896

119881119898+1

) 119902 (120574119895119896

1198811))

10038161003816100381610038161003816

=100381610038161003816100381610038161003816100381610038161003816int1198811(Ω)

int1198811(Ω)

119902 (120574119895119896

119909) 119902 (120574119895119896

119910)

times (119906(1198811 119881119898+1)

(119909 119910) minus 119906 (119909) 119906 (119910)) 119889119909119889119910100381610038161003816100381610038161003816100381610038161003816

le int1198811(Ω)

int1198811(Ω)

10038161003816100381610038161003816119902 (120574119895119896

119909)1003816100381610038161003816100381610038161003816100381610038161003816119902 (120574

119895119896 119910)

10038161003816100381610038161003816

times10038161003816100381610038161003816119906(1198811 119881119898+1)

(119909 119910) minus 119906 (119909) 119906 (119910)10038161003816100381610038161003816 119889119909 119889119910

le 119862(int1198811(Ω)

10038161003816100381610038161003816119902 (120574119895119896

119909)10038161003816100381610038161003816 119889119909)

2

le 119862221205881198952minus119895

(44)




and 120573119895119896


0be (9) and let 119895

1be (10)


0 119895


119895

E (10038161003816100381610038161003816119888119895119896 minus 119888

119895119896

100381610038161003816100381610038162

) le 11986222120588119895 1

119899 (45)

E (10038161003816100381610038161003816119889119895119896

minus 119889119895119896

100381610038161003816100381610038162

) le 11986222120588119895 1

119899 (46)


0 119895


119895

E (10038161003816100381610038161003816119889119895119896

minus 119889119895119896

100381610038161003816100381610038164

) le 11986224120588119895 (47)



1 and


P(10038161003816100381610038161003816119889119895119896

minus 119889119895119896

10038161003816100381610038161003816 ge120581120582

119895

2) le 119862

1

1198994 (48)



E (10038161003816100381610038161003816119888119895119896 minus 119888

119895119896

100381610038161003816100381610038162

) = V (119888119895119896

)

=1

1198992

119899

sum119894=1

V (119902 (120601119895119896

119881119894)) +

2

1198992

times119899

sumV=2

Vminus1

sumℓ=1

Re (C119900V (119902 (120601

119895119896 119881V) 119902 (120601

119895119896 119881

ℓ)))

=1

1198992

119899

sum119894=1

V (119902 (120601119895k 119881119894

)) +2

1198992

times119899minus1

sum119898=1

(119899 minus 119898)Re (C119900V (119902 (120601

119895119896 119881

119898+1) 119902 (120601

119895119896 119881

1)))

le1

119899(E (

10038161003816100381610038161003816119902 (120601119895119896

1198811)100381610038161003816100381610038162

)

+2119899minus1

sum119898=1

10038161003816100381610038161003816C119900V (119902 (120601119895119896

119881119898+1

) 119902 (120601119895119896

1198811))

10038161003816100381610038161003816)

(49)

By (H2)-(ii) we get

E (10038161003816100381610038161003816119902(120601119895119896

1198811)100381610038161003816100381610038162

) le 11986222120588119895 (50)


119899minus1

sum119898=1

10038161003816100381610038161003816C119900V (119902 (120601119895119896

119881119898+1

) 119902 (120601119895119896

1198811))

10038161003816100381610038161003816 = 119860 + 119861 (51)

where

119860 =[ln 119899120579]minus1

sum119898=1

10038161003816100381610038161003816C119900V (119902 (120601119895119896

119881119898+1

) 119902 (120601119895119896

1198811))

10038161003816100381610038161003816

119861 =119899minus1

sum119898=[(ln 119899)120579]

10038161003816100381610038161003816C119900V (119902 (120601119895119896

119881119898+1

) 119902 (120601119895119896

1198811))

10038161003816100381610038161003816

(52)


119860 le 119862221205881198952minus119895 [ln 119899

120579] le 11986222120588119895 (53)




119861 le 119862radicE (10038161003816100381610038161003816119902(120601119895119896

1198811)100381610038161003816100381610038164

)119899minus1

sum119898=[(ln 119899)120579]

radic120572119898

le 119862212058811989521198952radicE (10038161003816100381610038161003816119902(120601119895119896

1198811)100381610038161003816100381610038162

)infin

sum119898=[(ln 119899)120579]

119890minus1205791198982

= 11986222120588119895radic119899119890minus(ln 119899)2 le 11986222120588119895

(54)

Thus119899minus1

sum119898=1

10038161003816100381610038161003816C119900V (119902 (120601119895119896

119881119898+1

) 119902 (120601119895119896

1198811))

10038161003816100381610038161003816 le 11986222120588119895 (55)



| le

sup119909isin1198811(Ω)

|119902(120595119895119896


119895119896| le 119891

2le 119862 that

10038161003816100381610038161003816119889119895119896minus 119889

119895119896

10038161003816100381610038161003816 le10038161003816100381610038161003816119889119895119896

10038161003816100381610038161003816 +10038161003816100381610038161003816119889119895119896

10038161003816100381610038161003816 le 119862212058811989521198952 (56)


E (10038161003816100381610038161003816119889119895119896

minus 119889119895119896

100381610038161003816100381610038164

) le 119862221205881198952119895E (

10038161003816100381610038161003816119889119895119896minus 119889

119895119896

100381610038161003816100381610038162

) le 119862241205881198952119895 1

119899

(57)


Lemma 9 Let us set

119880119894= 119902 (120595

119895119896 119881

119894) minus E (119902 (120595

119895119896 119881

1)) (58)


119899(ln 119899)3

1003816100381610038161003816119880119894

1003816100381610038161003816 le 2 sup119909isin1198811(Ω)

10038161003816100381610038161003816119902 (120595119895119896

119909)10038161003816100381610038161003816 le 119862212058811989521198952 le 1198622120588119895

radic119899

(ln 119899)3

(59)



V (119897

sum119894=1

119880119894)

= V (119897

sum119894=1

119902 (120595119895119896

119881119894)) le 11986222120588119895 (119897 + 11989722minus119895) le 11986222120588119895119897

(60)

Therefore

119863119898

= max119897isin12119898

V (119897

sum119894=1

119880119894) le 11986222120588119895119898 (61)


119899 120582 = 120581120582

1198952


P(10038161003816100381610038161003816119889119895119896

minus 119889119895119896

10038161003816100381610038161003816 ge120581120582

119895

2)

le 119862(exp(minus11986212058121205822

119895119899

119863119898119898 + 120581120582

119895119898119872

) +119872

120582119895

119899119890minus120579119898)

le 119862( exp (minus119862((120581222120588119895 ln 119899)

times (22120588119895 + 1205812120588119895radic (ln 119899)

119899

times [radic120581 ln 119899] 2120588119895

radic119899

(ln 119899)3)

minus1

))

+ radic119899(ln 119899)3

(ln 119899) 119899119899119890minus120579[radic120581 ln 119899])

le 119862(119899minus1198621205812(1+120581

32) + 1198992minus120579radic120581)

(62)





119891 (119909) = sum119896isinΛ 1198950

1198881198950 119896

1206011198950 119896

(119909) +infin

sum119895=1198950

sum119896isinΛ 119895

119889119895119896

120595119895119896

(119909) (63)

where 1198881198950 119896

= int1

0119891(119909)120601

1198950 119896(119909)119889119909 and 119889

119895119896= int

1

0119891(119909)120595

119895119896(119909)119889119909


E (10038171003817100381710038171003817119891 minus 119891

100381710038171003817100381710038172

2) = 119875 + 119876 + 119877 (64)

where

119875 = sum119896isinΛ 1198950

E (100381610038161003816100381610038161198881198950 119896 minus 119888

1198950 119896

100381610038161003816100381610038162

)

119876 =

1198951

sum119895=1198950

sum119896isinΛ 119895

E(100381610038161003816100381610038161003816119889

1198951198961|119889119895119896|ge120581120582119895

minus 119889119895119896

100381610038161003816100381610038161003816

2

)

119877 =infin

sum119895=1198951+1

sum119896isinΛ 119895

1198892

119895119896

(65)




119875 le 11986221198950

119899le 119862

ln 119899

119899le 119862(

ln 119899

119899)

2119904(2119904+2120588+1)

(66)


119901119903(119872) sube 119861119904

2infin(119872)


119877 le 119862infin

sum119895=1198951+1

2minus2119895119904 le 119862((ln 119899)3

119899)

2119904(2120588+1)

le 119862(ln 119899

119899)

2119904(2119904+2120588+1)

(67)


119901119903(119872) sube

119861119904+12minus1119901



119877 le 119862infin

sum119895=1198951+1

2minus2119895(119904+12minus1119901)

le 119862((ln 119899)3

119899)

2(119904+12minus1119901)(2120588+1)

le 119862(ln 119899

119899)

2119904(2119904+2120588+1)

(68)


119877 le 119862(ln 119899

119899)

2119904(2119904+2120588+1)

(69)


= 119889119895119896

minus 119889119895119896


119876 =4

sum119894=1

119876119894 (70)

where

1198761=

1198951

sum119895=1198950

sum119896isinΛ 119895

E (10038161003816100381610038161003816119863119895119896

100381610038161003816100381610038162

1|119889119895119896|ge120581120582119895 |119889119895119896|lt1205811205821198952

)

1198762=

1198951

sum119895=1198950

sum119896isinΛ 119895

E (10038161003816100381610038161003816119863119895119896

100381610038161003816100381610038162

1|119889119895119896|ge120581120582119895 |119889119895119896|ge1205811205821198952

)

1198763=

1198951

sum119895=1198950

sum119896isinΛ 119895

E (1198892

1198951198961|119889119895119896|lt120581120582119895 |119889119895119896|ge2120581120582119895

)

1198764=

1198951

sum119895=1198950

sum119896isinΛ 119895

E (1198892

1198951198961|119889119895119896|lt120581120582119895 |119889119895119896|lt2120581120582119895

)

(71)



1|119889119895119896|lt120581120582119895 |119889119895119896|ge2120581120582119895

le 1|119895119896|gt1205811205821198952

1|119889119895119896|ge120581120582119895 |119889119895119896|lt1205811205821198952

le

1|119895119896|gt1205811205821198952

and 1|119889119895119896|lt120581120582119895 |119889119895119896|ge2120581120582119895

le 1|119889119895119896|le2|119895119896|


1198761+ 119876

3le 119862

1198951

sum119895=1198950

sum119896isinΛ 119895

E (10038161003816100381610038161003816119863119895119896

100381610038161003816100381610038162

1|119895119896|gt1205811205821198952

)

le 119862

1198951

sum119895=1198950

sum119896isinΛ 119895

(E (10038161003816100381610038161003816119863119895119896

100381610038161003816100381610038164

))12

(P(10038161003816100381610038161003816119863119895119896

10038161003816100381610038161003816 gt120581120582

119895

2))

12

le 1198621

1198992

1198951

sum119895=1198950

2119895(1+2120588) le 1198621

119899le 119862(

ln 119899

119899)

2119904(2119904+2120588+1)

(72)


Proposition 10 that

1198762le

1198951

sum119895=1198950

sum119896isinΛ 119895

E (10038161003816100381610038161003816119863119895119896

100381610038161003816100381610038162

) 1|119889119895119896|ge1205811205821198952

le 1198621

119899

1198951

sum119895=j0

22120588119895 sum119896isinΛ 119895

1|119889119895119896|gt1205811205821198952

(73)


2119895lowast = [(119899

ln 119899)

1(2119904+2120588+1)

] (74)


0 119895



1198762le 119876

21+ 119876

22 (75)

where

11987621

= 1198621

119899

119895lowast

sum119895=1198950

22120588119895 sum119896isinΛ 119895

1|119889119895119896|gt1205811205821198952

11987622

= 1198621

119899

1198951

sum119895=119895lowast+1

22120588119895 sum119896isinΛ 119895

1|119889119895119896|gt1205811205821198952

(76)


11987621

le 119862ln 119899

119899

119895lowast

sum119895=1198950

2119895(1+2120588) le 119862(ln 119899

119899)

2119904(2119904+2120588+1)

(77)




119901119903(119872) sube 119861119904


11987622

le 119862ln 119899

119899

1198951

sum119895=119895lowast+1

22120588119895 1

1205822

119895

sum119896isinΛ 119895

1198892

119895119896le 119862

infin

sum119895=119895lowast+1

sum119896isinΛ 119895

1198892

119895119896

le 119862infin

sum119895=119895lowast+1

2minus2119895119904 le 119862(ln 119899

119899)

2119904(2119904+2120588+1)

(78)


119901119903(119872) and (2119904 + 2120588 + 1)(2 minus 119901)2 +


11987622

le 119862ln 119899

119899

1198951

sum119895=119895lowast+1

22120588119895 1

120582119901

119895

sum119896isinΛ 119895

10038161003816100381610038161003816119889119895119896

10038161003816100381610038161003816119901

le 119862(ln 119899

119899)


sum119895=119895lowast+1


le 119862(ln 119899

119899)

(2minus119901)2


le 119862(ln 119899

119899)

2119904(2119904+2120588+1)

(79)


1198762le C(

ln 119899

119899)

2119904(2119904+2120588+1)

(80)


1198764le

1198951

sum119895=1198950

sum119896isinΛ 119895

1198892

1198951198961|119889119895119896|lt2120581120582119895

(81)


4can be bound as

1198764le 119876

41+ 119876

42 (82)

where

11987641

=

119895lowast

sum119895=1198950

sum119896isinΛ 119895

1198892

1198951198961|119889119895119896|lt2120581120582119895

11987642

=

1198951

sum119895=119895lowast+1

sum119896isinΛ 119895

1198892

1198951198961|119889119895119896|lt2120581120582119895

(83)


11987641

le 119862

119895lowast

sum119895=1198950

21198951205822

119895= 119862

ln 119899

119899

119895lowast

sum119895=1198950

2119895(1+2120588) le 119862(ln 119899

119899)

2119904(2119904+2120588+1)

(84)



119901119903(119872) sube 119861119904

2infin(119872)

we have

11987642

leinfin

sum119895=119895lowast+1

sum119896isinΛ 119895

1198892

119895119896le 119862

infin

sum119895=119895lowast+1

2minus2119895119904 le 119862(ln 119899

119899)

2119904(2119904+2120588+1)

(85)


119901119903(119872) and (2119904 + 2120588 + 1)(2 minus


11987642

le 119862

1198951

sum119895=119895lowast+1

1205822minus119901

119895sum

119896isinΛ 119895

10038161003816100381610038161003816119889119895119896

10038161003816100381610038161003816119901

= 119862(ln 119899

119899)

(2minus119901)2 1198951

sum119895=119895lowast+1

2119895120588(2minus119901) sum119896isinΛ 119895

10038161003816100381610038161003816119889119895119896

10038161003816100381610038161003816119901

le 119862(ln 119899

119899)


sum119895=119895lowast+1


le 119862(ln 119899

119899)

(2minus119901)2


le 119862(ln 119899

119899)

2119904(2119904+2120588+1)

(86)


1198764le 119862(

ln 119899

119899)

2119904(2119904+2120588+1)

(87)


119876 le 119862(ln 119899

119899)

2119904(2119904+2120588+1)

(88)





119891 (119909) minus 119891 (119909) = 119878 (119909) minus 119879 (119909) (89)


where

119878 (119909)

=1

119892 (119909)(V (119909) minus V (119909)

+119891 (119909) (119892 (119909) minus 119892 (119909))) 1|119892(119909)|ge119888lowast2

119879 (119909) = 119891 (119909) 1|119892(119909)|lt119888lowast2

(90)


|119878 (119909)| le 119862 (|V (119909) minus V (119909)| + 1003816100381610038161003816119892 (119909) minus 119892 (119909)1003816100381610038161003816) (91)





le 1198621003816100381610038161003816119892 (119909) minus 119892 (119909)

1003816100381610038161003816 (92)



1003816100381610038161003816) (93)


E (10038171003817100381710038171003817119891 minus 119891

100381710038171003817100381710038172

2) le 119862 (E (V minus V2

2) + E (

1003817100381710038171003817119892 minus 11989210038171003817100381710038172

2)) (94)




1are independent

with E(1205851) = 0

E (119902 (120574119895119896

1198811))

= E (1198841120574119895119896

(1198831)) = E (119891 (119883

1) 120574

119895119896(119883

1))

= int1

0

119891 (119909) 120574119895119896

(119909) 119892 (119909) 119889119909 = int1

0

V (119909) 120574119895119896

(119909) 119889119909

(95)

(H2) (i)-(ii)-(iii) with 120588 = 0


|1198841| le 119872 with 119872 gt 0 we have

sup(119909119909lowast)isin1198811(Ω)

10038161003816100381610038161003816119902 (120574119895119896

(119909 119909lowast))

10038161003816100381610038161003816


10038161003816100381610038161003816119909120574119895119896(119909

lowast)10038161003816100381610038161003816


10038161003816100381610038161003816120574119895119896(119909

lowast)10038161003816100381610038161003816 le 11986221198952

(96)


E (10038161003816100381610038161003816119902(120574119895119896

1198811)100381610038161003816100381610038162

) = E (1198842

1(120574

119895119896(119883

1))

2

)

le 119862E ((120574119895119896

(1198831))

2

)

= 119862int1

0

(120574119895119896

(119909))2

119892 (119909) 119889119909

le 119862int1

0

(120574119895119896

(119909))2

119889119909 le 119862

(97)



int1198811(Ω)

10038161003816100381610038161003816119902 (120574119895119896

119909)10038161003816100381610038161003816 119889119909

= (int119872

minus119872

|119909| 119889119909)(int1

0

10038161003816100381610038161003816120574119895119896(119909

lowast)10038161003816100381610038161003816 119889119909lowast

)

= 1198722 int1

0

10038161003816100381610038161003816120574119895119896(119909)

10038161003816100381610038161003816 119889119909 le 1198622minus1198952

(98)



119901119903(119872) with 119903 ge 1 119901 ge 2


E (V minus V22) le 119862(

ln 119899

119899)

2119904(2119904+1)

(99)


119905instead of 119881

119905


119901119903(119872) with 119903 ge 1 119901 ge 2


E (1003817100381710038171003817119892 minus 119892

10038171003817100381710038172

2) le 119862(

ln 119899

119899)

2119904(2119904+1)

(100)




Acknowledgment



References
































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119861 le 119862radicE (10038161003816100381610038161003816119902(120601119895119896

1198811)100381610038161003816100381610038164

)119899minus1

sum119898=[(ln 119899)120579]

radic120572119898

le 119862212058811989521198952radicE (10038161003816100381610038161003816119902(120601119895119896

1198811)100381610038161003816100381610038162

)infin

sum119898=[(ln 119899)120579]

119890minus1205791198982

= 11986222120588119895radic119899119890minus(ln 119899)2 le 11986222120588119895

(54)

Thus119899minus1

sum119898=1

10038161003816100381610038161003816C119900V (119902 (120601119895119896

119881119898+1

) 119902 (120601119895119896

1198811))

10038161003816100381610038161003816 le 11986222120588119895 (55)



| le

sup119909isin1198811(Ω)

|119902(120595119895119896


119895119896| le 119891

2le 119862 that

10038161003816100381610038161003816119889119895119896minus 119889

119895119896

10038161003816100381610038161003816 le10038161003816100381610038161003816119889119895119896

10038161003816100381610038161003816 +10038161003816100381610038161003816119889119895119896

10038161003816100381610038161003816 le 119862212058811989521198952 (56)


E (10038161003816100381610038161003816119889119895119896

minus 119889119895119896

100381610038161003816100381610038164

) le 119862221205881198952119895E (

10038161003816100381610038161003816119889119895119896minus 119889

119895119896

100381610038161003816100381610038162

) le 119862241205881198952119895 1

119899

(57)


Lemma 9 Let us set

119880119894= 119902 (120595

119895119896 119881

119894) minus E (119902 (120595

119895119896 119881

1)) (58)


119899(ln 119899)3

1003816100381610038161003816119880119894

1003816100381610038161003816 le 2 sup119909isin1198811(Ω)

10038161003816100381610038161003816119902 (120595119895119896

119909)10038161003816100381610038161003816 le 119862212058811989521198952 le 1198622120588119895

radic119899

(ln 119899)3

(59)



V (119897

sum119894=1

119880119894)

= V (119897

sum119894=1

119902 (120595119895119896

119881119894)) le 11986222120588119895 (119897 + 11989722minus119895) le 11986222120588119895119897

(60)

Therefore

119863119898

= max119897isin12119898

V (119897

sum119894=1

119880119894) le 11986222120588119895119898 (61)


119899 120582 = 120581120582

1198952


P(10038161003816100381610038161003816119889119895119896

minus 119889119895119896

10038161003816100381610038161003816 ge120581120582

119895

2)

le 119862(exp(minus11986212058121205822

119895119899

119863119898119898 + 120581120582

119895119898119872

) +119872

120582119895

119899119890minus120579119898)

le 119862( exp (minus119862((120581222120588119895 ln 119899)

times (22120588119895 + 1205812120588119895radic (ln 119899)

119899

times [radic120581 ln 119899] 2120588119895

radic119899

(ln 119899)3)

minus1

))

+ radic119899(ln 119899)3

(ln 119899) 119899119899119890minus120579[radic120581 ln 119899])

le 119862(119899minus1198621205812(1+120581

32) + 1198992minus120579radic120581)

(62)





119891 (119909) = sum119896isinΛ 1198950

1198881198950 119896

1206011198950 119896

(119909) +infin

sum119895=1198950

sum119896isinΛ 119895

119889119895119896

120595119895119896

(119909) (63)

where 1198881198950 119896

= int1

0119891(119909)120601

1198950 119896(119909)119889119909 and 119889

119895119896= int

1

0119891(119909)120595

119895119896(119909)119889119909


E (10038171003817100381710038171003817119891 minus 119891

100381710038171003817100381710038172

2) = 119875 + 119876 + 119877 (64)

where

119875 = sum119896isinΛ 1198950

E (100381610038161003816100381610038161198881198950 119896 minus 119888

1198950 119896

100381610038161003816100381610038162

)

119876 =

1198951

sum119895=1198950

sum119896isinΛ 119895

E(100381610038161003816100381610038161003816119889

1198951198961|119889119895119896|ge120581120582119895

minus 119889119895119896

100381610038161003816100381610038161003816

2

)

119877 =infin

sum119895=1198951+1

sum119896isinΛ 119895

1198892

119895119896

(65)




119875 le 11986221198950

119899le 119862

ln 119899

119899le 119862(

ln 119899

119899)

2119904(2119904+2120588+1)

(66)


119901119903(119872) sube 119861119904

2infin(119872)


119877 le 119862infin

sum119895=1198951+1

2minus2119895119904 le 119862((ln 119899)3

119899)

2119904(2120588+1)

le 119862(ln 119899

119899)

2119904(2119904+2120588+1)

(67)


119901119903(119872) sube

119861119904+12minus1119901



119877 le 119862infin

sum119895=1198951+1

2minus2119895(119904+12minus1119901)

le 119862((ln 119899)3

119899)

2(119904+12minus1119901)(2120588+1)

le 119862(ln 119899

119899)

2119904(2119904+2120588+1)

(68)


119877 le 119862(ln 119899

119899)

2119904(2119904+2120588+1)

(69)


= 119889119895119896

minus 119889119895119896


119876 =4

sum119894=1

119876119894 (70)

where

1198761=

1198951

sum119895=1198950

sum119896isinΛ 119895

E (10038161003816100381610038161003816119863119895119896

100381610038161003816100381610038162

1|119889119895119896|ge120581120582119895 |119889119895119896|lt1205811205821198952

)

1198762=

1198951

sum119895=1198950

sum119896isinΛ 119895

E (10038161003816100381610038161003816119863119895119896

100381610038161003816100381610038162

1|119889119895119896|ge120581120582119895 |119889119895119896|ge1205811205821198952

)

1198763=

1198951

sum119895=1198950

sum119896isinΛ 119895

E (1198892

1198951198961|119889119895119896|lt120581120582119895 |119889119895119896|ge2120581120582119895

)

1198764=

1198951

sum119895=1198950

sum119896isinΛ 119895

E (1198892

1198951198961|119889119895119896|lt120581120582119895 |119889119895119896|lt2120581120582119895

)

(71)



1|119889119895119896|lt120581120582119895 |119889119895119896|ge2120581120582119895

le 1|119895119896|gt1205811205821198952

1|119889119895119896|ge120581120582119895 |119889119895119896|lt1205811205821198952

le

1|119895119896|gt1205811205821198952

and 1|119889119895119896|lt120581120582119895 |119889119895119896|ge2120581120582119895

le 1|119889119895119896|le2|119895119896|


1198761+ 119876

3le 119862

1198951

sum119895=1198950

sum119896isinΛ 119895

E (10038161003816100381610038161003816119863119895119896

100381610038161003816100381610038162

1|119895119896|gt1205811205821198952

)

le 119862

1198951

sum119895=1198950

sum119896isinΛ 119895

(E (10038161003816100381610038161003816119863119895119896

100381610038161003816100381610038164

))12

(P(10038161003816100381610038161003816119863119895119896

10038161003816100381610038161003816 gt120581120582

119895

2))

12

le 1198621

1198992

1198951

sum119895=1198950

2119895(1+2120588) le 1198621

119899le 119862(

ln 119899

119899)

2119904(2119904+2120588+1)

(72)


Proposition 10 that

1198762le

1198951

sum119895=1198950

sum119896isinΛ 119895

E (10038161003816100381610038161003816119863119895119896

100381610038161003816100381610038162

) 1|119889119895119896|ge1205811205821198952

le 1198621

119899

1198951

sum119895=j0

22120588119895 sum119896isinΛ 119895

1|119889119895119896|gt1205811205821198952

(73)


2119895lowast = [(119899

ln 119899)

1(2119904+2120588+1)

] (74)


0 119895



1198762le 119876

21+ 119876

22 (75)

where

11987621

= 1198621

119899

119895lowast

sum119895=1198950

22120588119895 sum119896isinΛ 119895

1|119889119895119896|gt1205811205821198952

11987622

= 1198621

119899

1198951

sum119895=119895lowast+1

22120588119895 sum119896isinΛ 119895

1|119889119895119896|gt1205811205821198952

(76)


11987621

le 119862ln 119899

119899

119895lowast

sum119895=1198950

2119895(1+2120588) le 119862(ln 119899

119899)

2119904(2119904+2120588+1)

(77)




119901119903(119872) sube 119861119904


11987622

le 119862ln 119899

119899

1198951

sum119895=119895lowast+1

22120588119895 1

1205822

119895

sum119896isinΛ 119895

1198892

119895119896le 119862

infin

sum119895=119895lowast+1

sum119896isinΛ 119895

1198892

119895119896

le 119862infin

sum119895=119895lowast+1

2minus2119895119904 le 119862(ln 119899

119899)

2119904(2119904+2120588+1)

(78)


119901119903(119872) and (2119904 + 2120588 + 1)(2 minus 119901)2 +


11987622

le 119862ln 119899

119899

1198951

sum119895=119895lowast+1

22120588119895 1

120582119901

119895

sum119896isinΛ 119895

10038161003816100381610038161003816119889119895119896

10038161003816100381610038161003816119901

le 119862(ln 119899

119899)


sum119895=119895lowast+1


le 119862(ln 119899

119899)

(2minus119901)2


le 119862(ln 119899

119899)

2119904(2119904+2120588+1)

(79)


1198762le C(

ln 119899

119899)

2119904(2119904+2120588+1)

(80)


1198764le

1198951

sum119895=1198950

sum119896isinΛ 119895

1198892

1198951198961|119889119895119896|lt2120581120582119895

(81)


4can be bound as

1198764le 119876

41+ 119876

42 (82)

where

11987641

=

119895lowast

sum119895=1198950

sum119896isinΛ 119895

1198892

1198951198961|119889119895119896|lt2120581120582119895

11987642

=

1198951

sum119895=119895lowast+1

sum119896isinΛ 119895

1198892

1198951198961|119889119895119896|lt2120581120582119895

(83)


11987641

le 119862

119895lowast

sum119895=1198950

21198951205822

119895= 119862

ln 119899

119899

119895lowast

sum119895=1198950

2119895(1+2120588) le 119862(ln 119899

119899)

2119904(2119904+2120588+1)

(84)



119901119903(119872) sube 119861119904

2infin(119872)

we have

11987642

leinfin

sum119895=119895lowast+1

sum119896isinΛ 119895

1198892

119895119896le 119862

infin

sum119895=119895lowast+1

2minus2119895119904 le 119862(ln 119899

119899)

2119904(2119904+2120588+1)

(85)


119901119903(119872) and (2119904 + 2120588 + 1)(2 minus


11987642

le 119862

1198951

sum119895=119895lowast+1

1205822minus119901

119895sum

119896isinΛ 119895

10038161003816100381610038161003816119889119895119896

10038161003816100381610038161003816119901

= 119862(ln 119899

119899)

(2minus119901)2 1198951

sum119895=119895lowast+1

2119895120588(2minus119901) sum119896isinΛ 119895

10038161003816100381610038161003816119889119895119896

10038161003816100381610038161003816119901

le 119862(ln 119899

119899)


sum119895=119895lowast+1


le 119862(ln 119899

119899)

(2minus119901)2


le 119862(ln 119899

119899)

2119904(2119904+2120588+1)

(86)


1198764le 119862(

ln 119899

119899)

2119904(2119904+2120588+1)

(87)


119876 le 119862(ln 119899

119899)

2119904(2119904+2120588+1)

(88)





119891 (119909) minus 119891 (119909) = 119878 (119909) minus 119879 (119909) (89)


where

119878 (119909)

=1

119892 (119909)(V (119909) minus V (119909)

+119891 (119909) (119892 (119909) minus 119892 (119909))) 1|119892(119909)|ge119888lowast2

119879 (119909) = 119891 (119909) 1|119892(119909)|lt119888lowast2

(90)


|119878 (119909)| le 119862 (|V (119909) minus V (119909)| + 1003816100381610038161003816119892 (119909) minus 119892 (119909)1003816100381610038161003816) (91)





le 1198621003816100381610038161003816119892 (119909) minus 119892 (119909)

1003816100381610038161003816 (92)



1003816100381610038161003816) (93)


E (10038171003817100381710038171003817119891 minus 119891

100381710038171003817100381710038172

2) le 119862 (E (V minus V2

2) + E (

1003817100381710038171003817119892 minus 11989210038171003817100381710038172

2)) (94)




1are independent

with E(1205851) = 0

E (119902 (120574119895119896

1198811))

= E (1198841120574119895119896

(1198831)) = E (119891 (119883

1) 120574

119895119896(119883

1))

= int1

0

119891 (119909) 120574119895119896

(119909) 119892 (119909) 119889119909 = int1

0

V (119909) 120574119895119896

(119909) 119889119909

(95)

(H2) (i)-(ii)-(iii) with 120588 = 0


|1198841| le 119872 with 119872 gt 0 we have

sup(119909119909lowast)isin1198811(Ω)

10038161003816100381610038161003816119902 (120574119895119896

(119909 119909lowast))

10038161003816100381610038161003816


10038161003816100381610038161003816119909120574119895119896(119909

lowast)10038161003816100381610038161003816


10038161003816100381610038161003816120574119895119896(119909

lowast)10038161003816100381610038161003816 le 11986221198952

(96)


E (10038161003816100381610038161003816119902(120574119895119896

1198811)100381610038161003816100381610038162

) = E (1198842

1(120574

119895119896(119883

1))

2

)

le 119862E ((120574119895119896

(1198831))

2

)

= 119862int1

0

(120574119895119896

(119909))2

119892 (119909) 119889119909

le 119862int1

0

(120574119895119896

(119909))2

119889119909 le 119862

(97)



int1198811(Ω)

10038161003816100381610038161003816119902 (120574119895119896

119909)10038161003816100381610038161003816 119889119909

= (int119872

minus119872

|119909| 119889119909)(int1

0

10038161003816100381610038161003816120574119895119896(119909

lowast)10038161003816100381610038161003816 119889119909lowast

)

= 1198722 int1

0

10038161003816100381610038161003816120574119895119896(119909)

10038161003816100381610038161003816 119889119909 le 1198622minus1198952

(98)



119901119903(119872) with 119903 ge 1 119901 ge 2


E (V minus V22) le 119862(

ln 119899

119899)

2119904(2119904+1)

(99)


119905instead of 119881

119905


119901119903(119872) with 119903 ge 1 119901 ge 2


E (1003817100381710038171003817119892 minus 119892

10038171003817100381710038172

2) le 119862(

ln 119899

119899)

2119904(2119904+1)

(100)




Acknowledgment



References
































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119875 le 11986221198950

119899le 119862

ln 119899

119899le 119862(

ln 119899

119899)

2119904(2119904+2120588+1)

(66)


119901119903(119872) sube 119861119904

2infin(119872)


119877 le 119862infin

sum119895=1198951+1

2minus2119895119904 le 119862((ln 119899)3

119899)

2119904(2120588+1)

le 119862(ln 119899

119899)

2119904(2119904+2120588+1)

(67)


119901119903(119872) sube

119861119904+12minus1119901



119877 le 119862infin

sum119895=1198951+1

2minus2119895(119904+12minus1119901)

le 119862((ln 119899)3

119899)

2(119904+12minus1119901)(2120588+1)

le 119862(ln 119899

119899)

2119904(2119904+2120588+1)

(68)


119877 le 119862(ln 119899

119899)

2119904(2119904+2120588+1)

(69)


= 119889119895119896

minus 119889119895119896


119876 =4

sum119894=1

119876119894 (70)

where

1198761=

1198951

sum119895=1198950

sum119896isinΛ 119895

E (10038161003816100381610038161003816119863119895119896

100381610038161003816100381610038162

1|119889119895119896|ge120581120582119895 |119889119895119896|lt1205811205821198952

)

1198762=

1198951

sum119895=1198950

sum119896isinΛ 119895

E (10038161003816100381610038161003816119863119895119896

100381610038161003816100381610038162

1|119889119895119896|ge120581120582119895 |119889119895119896|ge1205811205821198952

)

1198763=

1198951

sum119895=1198950

sum119896isinΛ 119895

E (1198892

1198951198961|119889119895119896|lt120581120582119895 |119889119895119896|ge2120581120582119895

)

1198764=

1198951

sum119895=1198950

sum119896isinΛ 119895

E (1198892

1198951198961|119889119895119896|lt120581120582119895 |119889119895119896|lt2120581120582119895

)

(71)



1|119889119895119896|lt120581120582119895 |119889119895119896|ge2120581120582119895

le 1|119895119896|gt1205811205821198952

1|119889119895119896|ge120581120582119895 |119889119895119896|lt1205811205821198952

le

1|119895119896|gt1205811205821198952

and 1|119889119895119896|lt120581120582119895 |119889119895119896|ge2120581120582119895

le 1|119889119895119896|le2|119895119896|


1198761+ 119876

3le 119862

1198951

sum119895=1198950

sum119896isinΛ 119895

E (10038161003816100381610038161003816119863119895119896

100381610038161003816100381610038162

1|119895119896|gt1205811205821198952

)

le 119862

1198951

sum119895=1198950

sum119896isinΛ 119895

(E (10038161003816100381610038161003816119863119895119896

100381610038161003816100381610038164

))12

(P(10038161003816100381610038161003816119863119895119896

10038161003816100381610038161003816 gt120581120582

119895

2))

12

le 1198621

1198992

1198951

sum119895=1198950

2119895(1+2120588) le 1198621

119899le 119862(

ln 119899

119899)

2119904(2119904+2120588+1)

(72)


Proposition 10 that

1198762le

1198951

sum119895=1198950

sum119896isinΛ 119895

E (10038161003816100381610038161003816119863119895119896

100381610038161003816100381610038162

) 1|119889119895119896|ge1205811205821198952

le 1198621

119899

1198951

sum119895=j0

22120588119895 sum119896isinΛ 119895

1|119889119895119896|gt1205811205821198952

(73)


2119895lowast = [(119899

ln 119899)

1(2119904+2120588+1)

] (74)


0 119895



1198762le 119876

21+ 119876

22 (75)

where

11987621

= 1198621

119899

119895lowast

sum119895=1198950

22120588119895 sum119896isinΛ 119895

1|119889119895119896|gt1205811205821198952

11987622

= 1198621

119899

1198951

sum119895=119895lowast+1

22120588119895 sum119896isinΛ 119895

1|119889119895119896|gt1205811205821198952

(76)


11987621

le 119862ln 119899

119899

119895lowast

sum119895=1198950

2119895(1+2120588) le 119862(ln 119899

119899)

2119904(2119904+2120588+1)

(77)




119901119903(119872) sube 119861119904


11987622

le 119862ln 119899

119899

1198951

sum119895=119895lowast+1

22120588119895 1

1205822

119895

sum119896isinΛ 119895

1198892

119895119896le 119862

infin

sum119895=119895lowast+1

sum119896isinΛ 119895

1198892

119895119896

le 119862infin

sum119895=119895lowast+1

2minus2119895119904 le 119862(ln 119899

119899)

2119904(2119904+2120588+1)

(78)


119901119903(119872) and (2119904 + 2120588 + 1)(2 minus 119901)2 +


11987622

le 119862ln 119899

119899

1198951

sum119895=119895lowast+1

22120588119895 1

120582119901

119895

sum119896isinΛ 119895

10038161003816100381610038161003816119889119895119896

10038161003816100381610038161003816119901

le 119862(ln 119899

119899)


sum119895=119895lowast+1


le 119862(ln 119899

119899)

(2minus119901)2


le 119862(ln 119899

119899)

2119904(2119904+2120588+1)

(79)


1198762le C(

ln 119899

119899)

2119904(2119904+2120588+1)

(80)


1198764le

1198951

sum119895=1198950

sum119896isinΛ 119895

1198892

1198951198961|119889119895119896|lt2120581120582119895

(81)


4can be bound as

1198764le 119876

41+ 119876

42 (82)

where

11987641

=

119895lowast

sum119895=1198950

sum119896isinΛ 119895

1198892

1198951198961|119889119895119896|lt2120581120582119895

11987642

=

1198951

sum119895=119895lowast+1

sum119896isinΛ 119895

1198892

1198951198961|119889119895119896|lt2120581120582119895

(83)


11987641

le 119862

119895lowast

sum119895=1198950

21198951205822

119895= 119862

ln 119899

119899

119895lowast

sum119895=1198950

2119895(1+2120588) le 119862(ln 119899

119899)

2119904(2119904+2120588+1)

(84)



119901119903(119872) sube 119861119904

2infin(119872)

we have

11987642

leinfin

sum119895=119895lowast+1

sum119896isinΛ 119895

1198892

119895119896le 119862

infin

sum119895=119895lowast+1

2minus2119895119904 le 119862(ln 119899

119899)

2119904(2119904+2120588+1)

(85)


119901119903(119872) and (2119904 + 2120588 + 1)(2 minus


11987642

le 119862

1198951

sum119895=119895lowast+1

1205822minus119901

119895sum

119896isinΛ 119895

10038161003816100381610038161003816119889119895119896

10038161003816100381610038161003816119901

= 119862(ln 119899

119899)

(2minus119901)2 1198951

sum119895=119895lowast+1

2119895120588(2minus119901) sum119896isinΛ 119895

10038161003816100381610038161003816119889119895119896

10038161003816100381610038161003816119901

le 119862(ln 119899

119899)


sum119895=119895lowast+1


le 119862(ln 119899

119899)

(2minus119901)2


le 119862(ln 119899

119899)

2119904(2119904+2120588+1)

(86)


1198764le 119862(

ln 119899

119899)

2119904(2119904+2120588+1)

(87)


119876 le 119862(ln 119899

119899)

2119904(2119904+2120588+1)

(88)





119891 (119909) minus 119891 (119909) = 119878 (119909) minus 119879 (119909) (89)


where

119878 (119909)

=1

119892 (119909)(V (119909) minus V (119909)

+119891 (119909) (119892 (119909) minus 119892 (119909))) 1|119892(119909)|ge119888lowast2

119879 (119909) = 119891 (119909) 1|119892(119909)|lt119888lowast2

(90)


|119878 (119909)| le 119862 (|V (119909) minus V (119909)| + 1003816100381610038161003816119892 (119909) minus 119892 (119909)1003816100381610038161003816) (91)





le 1198621003816100381610038161003816119892 (119909) minus 119892 (119909)

1003816100381610038161003816 (92)



1003816100381610038161003816) (93)


E (10038171003817100381710038171003817119891 minus 119891

100381710038171003817100381710038172

2) le 119862 (E (V minus V2

2) + E (

1003817100381710038171003817119892 minus 11989210038171003817100381710038172

2)) (94)




1are independent

with E(1205851) = 0

E (119902 (120574119895119896

1198811))

= E (1198841120574119895119896

(1198831)) = E (119891 (119883

1) 120574

119895119896(119883

1))

= int1

0

119891 (119909) 120574119895119896

(119909) 119892 (119909) 119889119909 = int1

0

V (119909) 120574119895119896

(119909) 119889119909

(95)

(H2) (i)-(ii)-(iii) with 120588 = 0


|1198841| le 119872 with 119872 gt 0 we have

sup(119909119909lowast)isin1198811(Ω)

10038161003816100381610038161003816119902 (120574119895119896

(119909 119909lowast))

10038161003816100381610038161003816


10038161003816100381610038161003816119909120574119895119896(119909

lowast)10038161003816100381610038161003816


10038161003816100381610038161003816120574119895119896(119909

lowast)10038161003816100381610038161003816 le 11986221198952

(96)


E (10038161003816100381610038161003816119902(120574119895119896

1198811)100381610038161003816100381610038162

) = E (1198842

1(120574

119895119896(119883

1))

2

)

le 119862E ((120574119895119896

(1198831))

2

)

= 119862int1

0

(120574119895119896

(119909))2

119892 (119909) 119889119909

le 119862int1

0

(120574119895119896

(119909))2

119889119909 le 119862

(97)



int1198811(Ω)

10038161003816100381610038161003816119902 (120574119895119896

119909)10038161003816100381610038161003816 119889119909

= (int119872

minus119872

|119909| 119889119909)(int1

0

10038161003816100381610038161003816120574119895119896(119909

lowast)10038161003816100381610038161003816 119889119909lowast

)

= 1198722 int1

0

10038161003816100381610038161003816120574119895119896(119909)

10038161003816100381610038161003816 119889119909 le 1198622minus1198952

(98)



119901119903(119872) with 119903 ge 1 119901 ge 2


E (V minus V22) le 119862(

ln 119899

119899)

2119904(2119904+1)

(99)


119905instead of 119881

119905


119901119903(119872) with 119903 ge 1 119901 ge 2


E (1003817100381710038171003817119892 minus 119892

10038171003817100381710038172

2) le 119862(

ln 119899

119899)

2119904(2119904+1)

(100)




Acknowledgment



References
































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119901119903(119872) sube 119861119904


11987622

le 119862ln 119899

119899

1198951

sum119895=119895lowast+1

22120588119895 1

1205822

119895

sum119896isinΛ 119895

1198892

119895119896le 119862

infin

sum119895=119895lowast+1

sum119896isinΛ 119895

1198892

119895119896

le 119862infin

sum119895=119895lowast+1

2minus2119895119904 le 119862(ln 119899

119899)

2119904(2119904+2120588+1)

(78)


119901119903(119872) and (2119904 + 2120588 + 1)(2 minus 119901)2 +


11987622

le 119862ln 119899

119899

1198951

sum119895=119895lowast+1

22120588119895 1

120582119901

119895

sum119896isinΛ 119895

10038161003816100381610038161003816119889119895119896

10038161003816100381610038161003816119901

le 119862(ln 119899

119899)


sum119895=119895lowast+1


le 119862(ln 119899

119899)

(2minus119901)2


le 119862(ln 119899

119899)

2119904(2119904+2120588+1)

(79)


1198762le C(

ln 119899

119899)

2119904(2119904+2120588+1)

(80)


1198764le

1198951

sum119895=1198950

sum119896isinΛ 119895

1198892

1198951198961|119889119895119896|lt2120581120582119895

(81)


4can be bound as

1198764le 119876

41+ 119876

42 (82)

where

11987641

=

119895lowast

sum119895=1198950

sum119896isinΛ 119895

1198892

1198951198961|119889119895119896|lt2120581120582119895

11987642

=

1198951

sum119895=119895lowast+1

sum119896isinΛ 119895

1198892

1198951198961|119889119895119896|lt2120581120582119895

(83)


11987641

le 119862

119895lowast

sum119895=1198950

21198951205822

119895= 119862

ln 119899

119899

119895lowast

sum119895=1198950

2119895(1+2120588) le 119862(ln 119899

119899)

2119904(2119904+2120588+1)

(84)



119901119903(119872) sube 119861119904

2infin(119872)

we have

11987642

leinfin

sum119895=119895lowast+1

sum119896isinΛ 119895

1198892

119895119896le 119862

infin

sum119895=119895lowast+1

2minus2119895119904 le 119862(ln 119899

119899)

2119904(2119904+2120588+1)

(85)


119901119903(119872) and (2119904 + 2120588 + 1)(2 minus


11987642

le 119862

1198951

sum119895=119895lowast+1

1205822minus119901

119895sum

119896isinΛ 119895

10038161003816100381610038161003816119889119895119896

10038161003816100381610038161003816119901

= 119862(ln 119899

119899)

(2minus119901)2 1198951

sum119895=119895lowast+1

2119895120588(2minus119901) sum119896isinΛ 119895

10038161003816100381610038161003816119889119895119896

10038161003816100381610038161003816119901

le 119862(ln 119899

119899)


sum119895=119895lowast+1


le 119862(ln 119899

119899)

(2minus119901)2


le 119862(ln 119899

119899)

2119904(2119904+2120588+1)

(86)


1198764le 119862(

ln 119899

119899)

2119904(2119904+2120588+1)

(87)


119876 le 119862(ln 119899

119899)

2119904(2119904+2120588+1)

(88)





119891 (119909) minus 119891 (119909) = 119878 (119909) minus 119879 (119909) (89)


where

119878 (119909)

=1

119892 (119909)(V (119909) minus V (119909)

+119891 (119909) (119892 (119909) minus 119892 (119909))) 1|119892(119909)|ge119888lowast2

119879 (119909) = 119891 (119909) 1|119892(119909)|lt119888lowast2

(90)


|119878 (119909)| le 119862 (|V (119909) minus V (119909)| + 1003816100381610038161003816119892 (119909) minus 119892 (119909)1003816100381610038161003816) (91)





le 1198621003816100381610038161003816119892 (119909) minus 119892 (119909)

1003816100381610038161003816 (92)



1003816100381610038161003816) (93)


E (10038171003817100381710038171003817119891 minus 119891

100381710038171003817100381710038172

2) le 119862 (E (V minus V2

2) + E (

1003817100381710038171003817119892 minus 11989210038171003817100381710038172

2)) (94)




1are independent

with E(1205851) = 0

E (119902 (120574119895119896

1198811))

= E (1198841120574119895119896

(1198831)) = E (119891 (119883

1) 120574

119895119896(119883

1))

= int1

0

119891 (119909) 120574119895119896

(119909) 119892 (119909) 119889119909 = int1

0

V (119909) 120574119895119896

(119909) 119889119909

(95)

(H2) (i)-(ii)-(iii) with 120588 = 0


|1198841| le 119872 with 119872 gt 0 we have

sup(119909119909lowast)isin1198811(Ω)

10038161003816100381610038161003816119902 (120574119895119896

(119909 119909lowast))

10038161003816100381610038161003816


10038161003816100381610038161003816119909120574119895119896(119909

lowast)10038161003816100381610038161003816


10038161003816100381610038161003816120574119895119896(119909

lowast)10038161003816100381610038161003816 le 11986221198952

(96)


E (10038161003816100381610038161003816119902(120574119895119896

1198811)100381610038161003816100381610038162

) = E (1198842

1(120574

119895119896(119883

1))

2

)

le 119862E ((120574119895119896

(1198831))

2

)

= 119862int1

0

(120574119895119896

(119909))2

119892 (119909) 119889119909

le 119862int1

0

(120574119895119896

(119909))2

119889119909 le 119862

(97)



int1198811(Ω)

10038161003816100381610038161003816119902 (120574119895119896

119909)10038161003816100381610038161003816 119889119909

= (int119872

minus119872

|119909| 119889119909)(int1

0

10038161003816100381610038161003816120574119895119896(119909

lowast)10038161003816100381610038161003816 119889119909lowast

)

= 1198722 int1

0

10038161003816100381610038161003816120574119895119896(119909)

10038161003816100381610038161003816 119889119909 le 1198622minus1198952

(98)



119901119903(119872) with 119903 ge 1 119901 ge 2


E (V minus V22) le 119862(

ln 119899

119899)

2119904(2119904+1)

(99)


119905instead of 119881

119905


119901119903(119872) with 119903 ge 1 119901 ge 2


E (1003817100381710038171003817119892 minus 119892

10038171003817100381710038172

2) le 119862(

ln 119899

119899)

2119904(2119904+1)

(100)




Acknowledgment



References
































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










where

119878 (119909)

=1

119892 (119909)(V (119909) minus V (119909)

+119891 (119909) (119892 (119909) minus 119892 (119909))) 1|119892(119909)|ge119888lowast2

119879 (119909) = 119891 (119909) 1|119892(119909)|lt119888lowast2

(90)


|119878 (119909)| le 119862 (|V (119909) minus V (119909)| + 1003816100381610038161003816119892 (119909) minus 119892 (119909)1003816100381610038161003816) (91)





le 1198621003816100381610038161003816119892 (119909) minus 119892 (119909)

1003816100381610038161003816 (92)



1003816100381610038161003816) (93)


E (10038171003817100381710038171003817119891 minus 119891

100381710038171003817100381710038172

2) le 119862 (E (V minus V2

2) + E (

1003817100381710038171003817119892 minus 11989210038171003817100381710038172

2)) (94)




1are independent

with E(1205851) = 0

E (119902 (120574119895119896

1198811))

= E (1198841120574119895119896

(1198831)) = E (119891 (119883

1) 120574

119895119896(119883

1))

= int1

0

119891 (119909) 120574119895119896

(119909) 119892 (119909) 119889119909 = int1

0

V (119909) 120574119895119896

(119909) 119889119909

(95)

(H2) (i)-(ii)-(iii) with 120588 = 0


|1198841| le 119872 with 119872 gt 0 we have

sup(119909119909lowast)isin1198811(Ω)

10038161003816100381610038161003816119902 (120574119895119896

(119909 119909lowast))

10038161003816100381610038161003816


10038161003816100381610038161003816119909120574119895119896(119909

lowast)10038161003816100381610038161003816


10038161003816100381610038161003816120574119895119896(119909

lowast)10038161003816100381610038161003816 le 11986221198952

(96)


E (10038161003816100381610038161003816119902(120574119895119896

1198811)100381610038161003816100381610038162

) = E (1198842

1(120574

119895119896(119883

1))

2

)

le 119862E ((120574119895119896

(1198831))

2

)

= 119862int1

0

(120574119895119896

(119909))2

119892 (119909) 119889119909

le 119862int1

0

(120574119895119896

(119909))2

119889119909 le 119862

(97)



int1198811(Ω)

10038161003816100381610038161003816119902 (120574119895119896

119909)10038161003816100381610038161003816 119889119909

= (int119872

minus119872

|119909| 119889119909)(int1

0

10038161003816100381610038161003816120574119895119896(119909

lowast)10038161003816100381610038161003816 119889119909lowast

)

= 1198722 int1

0

10038161003816100381610038161003816120574119895119896(119909)

10038161003816100381610038161003816 119889119909 le 1198622minus1198952

(98)



119901119903(119872) with 119903 ge 1 119901 ge 2


E (V minus V22) le 119862(

ln 119899

119899)

2119904(2119904+1)

(99)


119905instead of 119881

119905


119901119903(119872) with 119903 ge 1 119901 ge 2


E (1003817100381710038171003817119892 minus 119892

10038171003817100381710038172

2) le 119862(

ln 119899

119899)

2119904(2119904+1)

(100)




Acknowledgment



References
































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










References
































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article A General Result on the Mean Integrated ...downloads.hindawi.com/journals/jps/2014/403764.pdf · Research Article A General Result on the Mean Integrated Squared