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No. 17958/1989
LIMIT THEOREMS FOR SUMS OF ORDER STATISTICS
SANDOR CSORGO*
This is a brief summary of recent results on the asymptotic distribution
of various ordered portions of sums of independent, identically distributed ran
dom variables that have been obtained by a direct probabilistic approach based
upon some integral representations of such sums in terms of uniform empirical
distribution functions, following a quantile transformation. Almost all the results
discussed here have been obtained jointly with Erich Haeusler and David M. Ma
son. The survey itself forms a guideline for a series of lectures given as a part of
the Sixth International Summer School.
1. Introduction
Let Xl, X 2 , ••• be independent, real, non-degenerate random variables with the
common distribution function F(x) = Pr{X· < x}, x E R, and introduce the inverse or
quantile function Q of F defined as
Q{s) = inf{x: F{x) > s}, 0 < s < 1, Q{O) = Q(O+).
Our motivating point of departure is the simple fact that if Ut, Ch, ... are independent
random variables on a probability space (0,.1", P) uniformly distributed in (0,1), then
for each n the distributional equality
(1.1)
•
holds, where Gn{s) = n-1#{1 < j < n : Uj ~ s} is the uniform empirical distribution
function on (0,1). In fact, if X1,n < ... < Xn,n are the order statistics based on the
* Partially supported by the Hungarian National Foundation for Scientific Research,
Grants 1808/86 and 457/88
1
sample Xl, ... , X n and Ul,n <Ub ... , Un' then we have
< Un,n denote the order statistics pertaining t.o
(1.2) {Xj,n: 1 <j< n, n > I} v {Q(Uj,n): 1 <j< n, n > I}
and hence, introducing the natural centering sequence
(1.3) [
l-(k+l)/n
Iln(m + l,n - (k + 1)) = 12 Q(u+)du,. (m+l)/n
we have, for example,
n-k n-k
L Xj,n - Jln(m + 1, n - (k + 1)) v L Q(Uj,n) - Jln(m + 1, n - (k + 1))j=m+l j=m+l
(1.4)
( [Um+J,n ( m+l) 1= ~ Q(Um+l,n) + n Gn(s) - dQ(s) ~
. (m+l)/n n
[
l-(k+l)/n
+ n (s - Gn(s))dQ(s). (m+l)/n
( ll-(k+l)/II ( n - k -1) 1+~Q(Un-k,71)+n , Gn(s)- n dQ(s)~
. L n-k-J,n
for any integers m, k > 0 such that m + 1 < 12 - k.
Relations (1.1) and (1.4) suggest the feasibility of a probabilistic approach to the
problem of the asymptotic distribution of sums of independent, identically distributed
random variables, or more generally, of the corresponding trimmed sums, to be based on
some properties of Q and the asymptotic behaviour of G n , rather than on characteristic
functions or other transforms of F. Kat,urally enough, the analytic conditions that this
method yields are all expressed in terms of the quantile function Q.
On the technical side, we are completely free to choose the underlying space
(O,:F, P). This will be the one described in [3,4]. It carries two independent sequences
{l'~j), n > I}, j = 1,2, of independent, exponentially distributed random variables
2
with mean 1 and a sequence {Bn(s), °~ s < 1; 71 ~ I} of Brownian bridges with the
properties that we describe now. For each n > 2, let
, j = 1, ... , [n/2],, j = [n/2] + 1, ... ,n + 1,
and for k = 1, ... , n + 1, ,,,-rite
k
Sk(n) = Llj(n).j=1
Then the ratios Uk,n = Sk(n)/Sn+l(n), k = 1, ... ,n, have the same joint distribution
as the order statistics of n independent uniform (0, 1) random variables, and for the
corresponding (left-continuous version of the) empirical distribution function
n
G(I)(s) = n-1~ I(U· < s) °< s < 1. n L..., J,n ,- - ,
j=1
where 1(·) is the indicator function, and the empirical quantile function
Un(s) = {[~k,n , (k - l)/n < s < ~'/n; k = 1, ... ,71,U1,n , S = 0,
we have
(1.5)
for any fixed v E [0,1/4) and
(1.6) In1/2 (s - Un(s» - Bn(s)1 0 ( -II)
sup = p nl/n~s~I-I/n (s(l - S»)1/2-11
for any fixed v E [0,1/2) as n ~ oc. ~Ioreover, the independent standard Poisson
processes00
Nj(t) = L1(Sij) < t), °<t < 00, j = 1,2,k=1
3
associated with the two independent jump-point sequences sij) = y1U) + ... +
l~j), j = 1,2, will be close enough for the present purposes to the random functions
nG<.j)(t/n), j = 1,2, respectively as n -+ 00, where
n
G~2)(s) = n-1 L 1(1 - Un+1-i,n < s), 0 ~ s < 1,j=l
is the empirical distribution function obtained by "counting down" from 1.
The quantile-transform method based on (1.2) has long been in use in statistical
theory and scattered applications of it can be found also in probability. Here we don't
aim at giving any bibliography of this method, a good source for its earlier use is the
book [53]. It is the approximation results in (1.5) and (1.6) in combination with Poisson
approximation techniques for extremes that has made this old method especially feasible
for the handling of problems of the asymptotic distribution of various sums of order
statistics.
The approach touched upon above was first used in [3] and [4] to obtain probabilistic
proofs of the sufficiency parts of the normal and stable convergence criteria, respectively,
for whole sums Ej=l Xj. The effect on the asymptotic distribution of tril1uning off a efixed number m of the smallest and and a fixed number k of the largest summands. Le.
the investigation of the lightly trimmed sums Tn(m, h') = Ej;:~+l Xj," , was already
considered in [4] under the (quantile equivalent of the) classical stable convergence crite-
rion. This line of research goes back to Darling [19] and Arov and Bobrov [1], with later
contributions by Hall [36], Teugels [55], 1Ialler [43]. ~Iori [49], Egorov [21] and Yino
gradov and Godovan'chuk [56]. (Again, we don't intend to compile full bibliographies
of the problems considered.) The earlier literature is concentrated almost exclusively on
trimmed sums where summands with largest absolute values are discarded. \Ye shall re-
fer to this kind of trimming as modulus or magnitude trimming in the sequel, as opposed
to our natural-order, or simply natural, trimming described above.
The paper [13] has initiated the study of two problems. One ,vas the problem of
the asymptotic distribution of moderately trimmed sums Tn(k'l' kn), where kn -+ ex: as
4 e
n -+ 00 such that kn/n -+ 0, the other one was the same problem for the corresponding
extreme sums Tn(O, n - kn) and Tn(n - kn,0). The first problem was looked at under the
restrictive initial assumption that F belonged to the domain of attraction of a normal
or a non-normal stable distribution, while the second one only in the non-normal stable
domain. Lat~r, the second problem concerning extreme sums 'was solved in [15] for all
F with' regularly varying tails and, extending a result in [14]; Lo [40] determined the
asymptotic distribution of extreme sums for all F which are in the domain of attraction
of a Gumbel distribution in the sense of extreme value theory. All these papers use the
probabilistic method.
The method itself has been perfected in the three papers [9,10,11], where a general
pattern of necessity proofs has also been worked out, which together constitute a general
unified theory of the asymptotic distribution of sums of order statistics. The next three
sections are devoted to a very brief sketch of this theory according to [10L [9] and [11],
respectively. Some related matters are taken up even more briefly in Section 5.
In these lectures vv'e shall only deal with problems in probability. The same method
has already been used to tackle some problems in asymptotic statistics. Some of these
applications, where the mathematics is most closely related to things discussed here! can
be found in [5], [8], [17] and [18].
An earlier and much shorter survey of the method is in [16]. In comparison~ the
present survey may be looked upon as a progress report covering the last two years.
2. Full and lightly trimmed sums [10]
The aim is. to determine all possible limiting distributions of the suitably centered
and normalized sequencen-k
Tn(m, k) = I: .Yj .n ,
j=m+l
where m ~ 0 and k > 0 are fixed, along subsequences of {n} under the broadest possible
conditions.
5
Choose the integers 1and l' such that m ~ 1~ l' ~ n - l' ~ n -I < n - h', and write
Tn(m,k)-Pn{m+1,n-(k+1))= {.t X j ,n-1/n(m+l,1+ll})=»1+1
+ { t Xi,n - Pn(l + 1, r + I)}i=/+1
+ { I: ){i,n - /In{1' + 1, n - (1' + In}j=r+l
{
71-1 }+ L X i ,n-/ln{n-(r+1),n-(/+1))
i=n-r+l
+{ I: X i ,n-J1n{n-(/+l),n-{k+1))}j=71-/+1
= v~)(l, 11) +.61(1,1', n) + m(r, 11)+
(2) 1 )+ 62 (l, r, 11 ) + t'k (, 11. ,
Now if we introduce
and
t,,(2){S) = ( -1.. Q (1-;)Tn ~-l.. Q{an)
,O<s<n-nan ,
, 11 - 1H} 71 < s < 00,
, 0 < s < 11 - 110'71,
, 11 - nan < s < 00,
,j = 1,,j = 2,
where An > 0 is some potential normalizing sequence and an -+ 0 as 11 -+ 00 such that
nan -+ 0 (so that P{O'n < U1 ,n < Un,n < 1- a,J -+ 1 as 11 -+ (0), and also
ZU) _ { nUq,nq,n - n(l - Un+1- q,n)
then, using (1.2) and integration by parts, for
lt1 i ){/, n) =171
vii> (I, n), h = m, k; j = 1,.2,
6
(2.2)
we can write
IT we now assume that there exists a subsequence {n'} of the positive integers such that
for two non-decreasing, right-continuous functions 'PI and 'P2 we have
(2.1) <p~)(s) ~ <pj(s), as n' ~ 00, at every ~ontinuity point s E (0,00) of Yj. j = 1. 2,
then it turns out that the right-side of the last distributional limit converges in proba
bility to a limit as n' ~ 00 for each fixed 1, and these limits converge, if we let 1 ......,. x,
in probability to
h = m, k; i =1,2. These limits are well-defined random variables because condition
(2.1) implies that
.1.00
'P](s)ds < 00 for any e > 0, j =1,2.
Also, it turns out that the terms 6j (l, 7', n'), j = 1,2, above only play the role of
"sanitary cordons" in the sense that under (2.1), 6j (1, r, n')/Anl converge to some limits
7
in probability as n' ~ 00, j = 1,2, and if we let 1~ 00 (forcing r ~ 00) then both
of these limit sequences converge to zero in probability. This fact shows that these two
strips do not contribute to the limit and they only separate ~~1) and l,~2) from a possibly
vanishing normal component of the limit coming from the middle term
Af(r, n') = Al m(r, n'),II'
which component by later appropriate choices of I = In' and r = rn' and by an applica
tion of a result of Rossberg [52] will be independent of the vector (l/~), VP)), the h ..·o
components of the latter being independent by construction.
Finally, using (1.2) and the representation (1.4) with m = k = r, and (1.5), it can
be shown that for any sequence rn' ~ 00, rn' In' ~ 0, we have
as n' ~ 00, where0< u , = U«7' n ' + 1)/71'} < 1
- n u(l/n')-
and an' = ';:;;;u(l/n'), where for 0 < s < 1,
[
1-S [1-S
u2 (s) =. S .Is (min(u,v) - ul')dQ(u)dQ(v)
and where
ll-(r..,+I)/n' /
Nn,(O, 1) = Bn,(s)dQ(s) u((rn, + 1)/n'). (r..,+I)/n' .
is a standard normal (lV(O, 1)) random variable for each n'.
This is the way we arrive at the direct half (i) of the following result which comprises
the essential elements of Theorems 1-5 in [10].
RESULT. (i) Assume (2.1) and that
(2.3)
8
where 6 is some non-negative constant. If 6 = 0, tlIen
A1,{ .~k Xj,n' _ !,n.(m + 1, n' - (k + I»)} ~ V~l) + V~2)
n ]=m+l
as n' ~ 00, where, necessarily, <pj(s) = 0 if S > 1 j = 1,2. If 6 > 0, tllen for any
subsequence {n"} of {n'l for "dlic1l Un" ~ U as n" ~ 00, wllere 0 ~ U < 1, 'we hare
{
n"-k }AI" L Xj,n" - Il nl/(m + 1, nil - (k + 1)) ~ V~l) +6cnV(0, 1) + V~~)
n j=m+l
as nil ~ 00, where the three terms ill the limit are independent. In botll cases l·~l) is
non-degenerate if <PI ¢ 0 and VP) is non-degenerate if <P2 ¢ O.
(ii) If there exist tn'o sequences of constants An > 0 and Cn and a sequence {n /} of
positive integers such that
(2.4){
n'-k }
A1
n, . L .'lj,n'"- Cn,
]=m+l
e converges in distribution to a non-degellerate limit, then there exist a subsequence {nil}
of {n' } and non-decreasing, non-positive, rigllt-continuous functions <PI and <P2 defined
on (0,00) satisfying (2.2) and a constant 0 < 6 < 00 sudl tbat (2.1) and (2.3) hold true
for An" along {nil}. The limiting random raria1Jle of tIle sequence in (2.4) is necessarily
of the form V~I) +6u2V(O, 1) +VP) + d lritll independent terms, 'VI'here
(2.5) d = lim dn", = lim {Jln",(m + 1, n'" - (k + 1)) - Cn", }/an",n''' ..... oo n'''-oo
for some subsequence {n"'} of {nil}. If 6 > 0 then eitlH~r u > 0 or at least one of <PI and
<P2 is not identically zero. If 6 = 0 tllen <Pj = 0 011 [1,(0), j = 1,2, but at least one of
them is not identically zero.
(2.6)
In the proof of the converse half (ii), the case when
A, (.) .lim sup _n l<p;, (s)1 < 00, 0 < S < 00;n' .....~ an'
9
j = 1,2,
is trivial, for then by Helly-Bray selection and the converg{'nce of types theorem there
exist a subsequence {n"'} such that (2.1), (2.3) and (2.4) all hold along it and 6 > 0 in e(2.3), and we can apply the direct half with a ~ o.
\Vhen, contrary to (2.6), there exists {n"} C {n'} such that
(2.7) li An" (I)()_m 'P n ll S --00n"_1X:; an"
for some s > 0, for which one can show that necessarily s < 1, then the sequence in (2.4)
is equal in distribution to
(2.8)
where Rhl) , ~Vn and Rh2
) result from dividing by an' the three terms on the right- side
of (1.4), respectively. Then again Rossberg's [52] result implies that the two sequences
IR~I)I and IR~2)1 are asymptotically independent, and we can show that
lim liminfP{IR~PI < AI} > 0, j = 1,2,AI -IX:; n-oo
which is somewhat less than the stochastic boundedness of the sequences of RCj), j =
1,2. However, it can be shown that these two facts and the stochastic boundedness of
the sequence in (2.8) (which holds since by assymption it has a limiting distribution)
already imply that both sequences
(2.9)D~) = Hn'IR~)1 = an'IR~)l/max(an"An')'
j = 1,2, are stochastically bounded.
However, on the event {Um+I,n" < sIn"} with positiv limiting probability of P{Sm+1 <s}, where s is as in (2.7), we have
(I) (S) An" (I)()Dn" > IQ ,,+ II max(a n ", An") = Hn" --'Pn" Sn an"
10
because the integral term in R~/) is non-positin~ for large enough 11 and Q(sjn") is non
positive for large enough n" (otherwise (2.7) could not happen). This fact, (2.7) and
(2.9) imply that an" IAn" --+ 0 as n" --+ 00 and that
limsupl<p~l,?(s)1< 0<:,,0 < s < ex).n"-oo
By repeating this proof if necessary one can choose a further subsubsequence {n"'} C
{n"} to arrive at
lim sup l<p~~?, (s) I < ':)0, 0 < s < 00,n"'-oo
and hence by a final application of a Helly-Bray selection we are done again.
Noting that the integral term in R~;) is non-negative for large enough 11, the subcase
when (2.6) fails for j = 2 is entirelly analogous.
The special case m = k = O·of the result a.hove gives an equivalent version of the
classical theory of the asymptotic distribution of independent, identically distributed
random variables (see, e.g. [25]) with a condition formulated in terms of the quantile
function. In this case, the limiting random variable in the direct half (i) is in general
~ro,o = VO(l) + pN(O, 1) + l~2), where p = 60 '2: 0 and
for j = 1,2. This is an infinitely divisible random variable with characteristic function
(2.10)
(0 ( 'f ). 1 ?? . z·x
EeztVo,o = exp it'}' - -p-t- + r e ztz - 1 - dL(x)2 .I-x 1 + x 2
+ lX (eitz _ 1 _ iiX?) dR(x)l,1+x-. 0
11
t E R, where "I = "11 + ')'2 with
. - (_ )j+1 ([1 CPj(S) d l~ 9~(S) 1._"I] - 1 ~ 1 + ~() S - 1 + ~() ds ~ , J - 1,2,
. 0 cpJ S . 1 9] S
and L(x) = inf{s > 0: 91(S) > x}, -00 < :r < 0, and R(J:) = inf{s < 0: -Y2(-S) >x}, 0 < x < 00, and any infinitely divisible random variable can be represented as lo.o
plus a constant (Theorem 3 in [10]) by reversing the definitions of the inverse functions
if a pair (L, R) of left and right Levy measures is given. (See Section 5.3 below.) So the
result above shows rather directly how these measures arise, while the sketched proof
indicates which portions of the whole sum contribute these extreme parts of the limiting
infinitely divisible law.
The direct and converse halfs of the result above are used in [10] to derive necessary
and sufficient conditions for full or lightly trimmed SUIns to be in the domain of attraction
of a normal law (the normal convergence criterion) or to be in the domain of partial
attraction of a normal law, for full sums to be in the domain of attraction of a non-normal
stable law (the stable convergence criterion; stable laws of exponent 0 < Q < 2 arise with
the functions CPj(s) = -CiS-I/o, 0 < s < oc., j = 1,2, where e1, C2 > 0 are constants
such that C1 +C2 > 0) or to be in the domain of partial attraction of a non-normal stable
law. The domains of normal attraction of these laws are also characterized. Analogous
characterization results are derived for the domain of partial attraction of some infinitely
divisible law or its lightly trimmed version V~l.k = l~l) + pJY(O, 1) + l~2), and necessary
and sufficient conditions are derived for the stochastic compactness and subsequential
compactness of lightly triIIUl1ed or full sums, together with a Pruitt-type [50] quantile
description of the arising subsequential limiting laws in the compact case. AH these
results are deduced from (i) and (ii) above, independently of the existing literature. all
the obtained necessary and sufficient conditions are expressed in terms of the quantile
function and hence are of independent interest, and most of the results are effectively
new as far as light trimming is concerned.
12
3. Moderately and heavily trimmed sums [9]
\Ve call the sumn-1..·n
Tn = Tn(m ll , kn ) = L Xj,nj=m n +l
moderately trimmed if the integers m n and kn are such that, as n --+ 00,
(3.1) m n --+ 00, kn --+ 00, mn/n --+ 0, kn/n --+ O.
Now, fixing these two sequences {m n} and {It'n}, with
[
1-1..'n /nJ1n = J1n(mn, n - kn) = n Q(s)ds
. mnln
the equality (1.4) simplifies to
(3.2)
1 v n [Um n•n
( m )T{Tn - JIn} = A Gn(s) - ~ dQ(s)n n . m n In _ n
n [1-knln+ An . mnln (s - Gn{s))dQ{s)
+~ /,l-knln (Gn(S) _ n - kn ) dQ(s)An . Un-h.n n
where An > 0 is some potential norming sequence, and R 1•n < 0, R 2 ,n > O.,
First, using (1.5) it turns out that
Yn = ~: (Zn +op(1)),
where now an is defined to be an = Vna(mn/n, 1- h'n/n), where for 0 < s $ t < 1.
(3.3)
and where
0 2(8,1) = [[(min(u,c) - ut')dQ(u)dQ(c),
1 [l-kn InZn = - Bn(s)dQ(s)
a{mn/n,1 - It'n/n) . mn/n .
13
is a standard normal variable for each n.
Concerning RI,n, it is easy to see from (1.6) that
7/2 (Umn,n - mn
) + ZI,n = Op(m;;l.I)mn n
for any 0 < v < 1/4, where ZI,n = (n/m n)I/2 Bn(mn/n). Using this in conjunction with
(1.5), it can be shown that RI,n behaves asymptotically as
which in turn behaves asymptotically as
.fv-Z'.. (ZI.n +x)d¢~l)(x) . .{Z". ~,!,l)(x)dx,
provided the sequence of functions
(1) ( m1
/2
)1j.'n -~
1/2 {( 1/2) }"ln Q r:n + X m~ - Q (":t )(1) (m1
/2
)tPn ~
m 1/ 2
,-00 <x < -~,
I I m~/2, .1: < -2-'
1/2,~<x < 00,
is at least bounded. Similarly, it can be shown that R2,n behaves asymptotically as
fO l-Z2.n,_ (Z2,n + X)d~·~2)(x) = 1j..<;;) (x)dx,. -Z2,n ·0
where Z2,n = (n/kn)I/2B n(1- kn/n) and
.1 (2) ( k1
/2
)'f'n -T1:2 {Q (1 - ~ + X~·~2) - Q (1 - ~)}
tP~2) (k~2)
14
k1 /2-00 < X < -T'
k 1 /2,lxl<T'
k 1 /2'T < X < oc.
For each n, (ZI,n, Zn, Z2,n) is a trivariate normal vector with covariance matrix
(
1 - .!!!..n.n Tl,n
Tl,n 1
( mnn2kn )1/2 T2,1l
where
( m)1/2 (m )1/2l1-knI1l / ( 1. )- 1--2. <Tl,n=- _n (l-s)dQ(s) a nnln,l_hn~n :::;0n n. mnln
and
( k )1/2 (k )1/2l1-kn/ll- 1 - --!!. < T2,n = - --!!. ..' sdQ(s)
n n. Tnnln
IT we brake up Zn as
then
El' r2 _ 2 _ 2 (m 11 1) / 2 (m11 1 kn )VI - al - a - - a - --,n ,n n ' 2 n ' n'
EW.2 ? 2 (1 kn) / ? (mn kn)2 =a2- =a -,1-- a- -,1-- ,,n ,n 2 n n n
where ar,n +aln= 1 for each n, and it can be shmvn that if EX2 = 00 then the three
covariances COV(ZI,n, H'2,n), Cov(lV1,n, H'2,n) and COV(Z2,n, lV1,n) all converge to zero
as n --. 00.
This is the way we arrive at the direct half (i) of the following main result of [9],
where this result is formulated somewhat differently. The proof of the converse half (ii)
15
go.es along the same line as that of the proof of the converse half in the preceding section,
the last step being technically different but the same in spirit. eRESULT. (i) Assume tllat there exists a subsequellce {n'} of tIle positive integers
such that for two non-decreasing, left-continuous functions~'l and tI'2 satis(ving ~'j(O) <0, tPj(O+) > 0, j = 1,2 lre have
(3.4) 1f'~)(x) -+ 'l/Jj(x), at every continuity point x E R of 'l/Jj, j = 1,2,
and that
(3.5) an,/An, -+ 6 < 00,
where an = n l / 2a(mn/n, 1- h'n/n) and 6 is some non-negati,'e constant. If 6 = 0, then
necessarily th(x) = 'tI-'2( -x) = 0 for all x > 0, alld, with lin = J111(mn, n - h'n} given
above
where
V; = V;( .pi) = (_1)i+1 lz ~'i(x)dx, j = 1,2,J
where Zl and Z2 are independent standard normal ralldom varial)les. If6 > 0, tllen for
any subsequence {n"} of in'} for whidl 7·j.n" -+ 7'j, j = 1,2, ",'here -1 < 7'1, r2 < O. we
necessarily have th(x) < -rl and 1f'2(X} > r2 for all x E R, and
where, with Zl and Z2 figuring in VI and V2, (Zll Z, Z2) is s trivariate normal random
vector with zero mean and covariance matrix
(~7'1
1
16
~) . .'
Aloreover, ifVar(X) = 00 and, If necessary, {n lll} is a furtller subsequellce of {nil} sudl
that for some positive constants 0'1 and 0'2 \l'itll ai +a~ = 1. aj,n '" ~ aj, j = 1,2, tllen
where (Zl' WI, TV2, Z2) is a quadrivariate normal vector ll"ith meall zero and co"'ariance
matrix
(ii) If there exist two sequences of constants An > 0 and en and a sequence {n /} of
positive integers such that
(3.6)
e converges in distribution to a non-degenerate limit, then tllere exist a subsequence {nil}
of {n /} and non-decreasing, left-colltinuous functions ~'l and l'2 satisfying ~'j(O) < 0 and
tI'j(O+) > 0, j = 1,2, and a constant 0 < b < 00 SUdl tllat (3.4) and (3.5) llOld true for
An" along {nil}, where at least one of ~'l and ~'2 is not identically zero if6 = 0, in Wllidl
case th (x) = ~'2(-x) = 0 for all x > O. The limiting random variable of the sequence
in (3.6) is necessarily of the form VI + bZ + l2 + d, with Fl. Z and \'2 described above,
where
d = lim (/ln lll - Cnlll)/An IIIn"l-oo
for some subsequence {n"'} C {nil}.
This result implies as a corollary (by showing that the possible limits can only be
normal if tPl =0 =~'2) that the sequence in (3.6) converges in distribution to a standard
normal random variable if and only if <p~)(x) ~ 0, as n' ~ Xi, for all x E R, j = 1,2.
17
In this case An' can be chosen to be an' and C n, to be Ji",. Here {n /} is arbitrary and
can of course be {n}. eFor a discussion of the conditions (3..1) and (3.5) we refer to the original paper [9]
and [31]. Subsequent to [9], Griffin and Pruitt [27] used the more classical characteristic
function methodology to prove mathematically equivalent versions of the above results
and showed that all possible subsequential limits indeed arise.. In another paper, [26],
they deal with the analogous problem of moderately trimmed sums when 1.."n(kn --+
00, kn/n --+ 0) of the summands largest in absolute value are discarded at each step
(see also [51]), assuming that the underlying distribution is symmetric about zero. (As
a demonstration of the present approach, a part of their results is rederived in [12].) A
~omparison of the two sets of results shO\vs that (perhaps contrary to intuition) even
if we assume symmetry and mk = 1.."n above, the two trimming problems are wholly
different. For further results on various interesting versions of moderate trimming see
Kuelbs and Ledoux [38], Hahn, Kuelbs and San~ur [33], Hahn and Kuelbs [32], Hahn,
Kuelbs and \Yeiner [34,35] and ~Ialler [44].
Finally, we turn to heavy trimming assuming instead of (3.1) that e(3.7) mn = [11Q] and kn = n - [;3n], 0 < 0' < (3 < 1.
This is the case of the classical trimmed sum. for which Stigler [54] completely soh'ed
the problem of asymptotic distribution. Suppose that 0'(a, (3) > 0, '''''here 0'(.,.) is as
in (3.3). The proof sketched above produces the following version of Stigler's theorem
(Theorem 5 in [9]), where an and Iln are defined in terms of the present m n and 1..~n: For
any underlying distribution,
as n --+ 00, where
.i'! (x) = {:(~~) (Q{a+) - Q{a))
18
, x <0,, x > 0,
and
lh(x) = f8(Q(,3+) - Q(8))(1(0,;3) I .
, x < 0,
, x> 0,
so that
and
VI(';!'I) = U(-;{3) (Q(a+) - Q(a)) min (0, ZJl
where (ZI, Z, Z2) is a trivariate normal random vector with mean zero and covariance
matrix
(
1- 0
Tl
(0'(1 - ,3))1/2
,
where{3 . {3
Tl = -..;G.fa (1 - s)dQ(s) and 7'2- = -vI -;3.fa sdQ(s),
e This version puts Stigler's theorem (giving asymptotic normality if and only if Q is
continuous both at 0 and ,3) into a broader picture. Substituting a and ,3 for mn/n and
1 - kn/n in the arguments of ~,~1) and 'lj.,~2), the proof also works for more general Inn
and kn sequences, provided y'n(mn/n - 0) -... 0 and y'n(1- kn/n - ,3) -... 0 as n -... 00.
For rates of convergence in Stigler's theorem in the case when the limit is normal,
see Egorov and Nevzorov [22], who in [23] also investigate the related problem in the
case of magnitude trimming,
4. Extreme sums [11]
Here we are interested in the sums of extreme values
k n n
En = En(kn) = L X n+1"':'j,n = L Xj,n,j=1 j=n-kn +l
19
where kn --+ 00 and either kn/n --+ 0 as n --+ 00, or ~'n = [no] with 0 < Q < 1. ('Ye shall
refer to the first case when J..·n/n --+ 0 as the case a = 0.) It is more convenient to work ewith the function
H(s)=-Q((l-s)-),O<s<1.
instead of Q itself, for \vhich we have
(Xl,n, ... , Xn,n) V (-H(Un,n),"" -H(Ul,n))
instead of (1.2). The natural centering sequence now turns out to be
/,
knln (1)Pn = J1n(~'n) = -n H(s)ds - H - ,
. lIn n
and with a potential normalizing sequence All > 0 the role of (1.4) or (3.2) is taken over
by the decomposition
and
The numbers mn and In (not necessarily integers) will be appropriately chosen such thatA (1) •
mn --+ 00, In/rnn --+ 00 and kn/ln --+ 00 as n --+ 00, and we see that the term Un IS
20
•
very similar to VJ1)(l, n) in Section 2, while the t.erm ~~) presents a "trimmed-sum
problem" considered in Section 4 with only one Rn-like term. Therefore, we need both
<pn and 1/.'n type functions, which are presently defined as
, 0 < s ~ n - nan,
, n - nO'n < s < 00,
where an is as in Section 2, and
I ( k~/2)'lI-'n --2-
k1
/2 {(k k
1/
2) (I..' )}-"- H =. + X-"- - H =.An n n n
(k
J/
2)'l/.'n T
k 1 / 2
- 00 < X < _::.::..!l-, 2 '
k 1 / 2
,1.1'1 < T'k J / 2
'T < X < 00.
Again, the middle term ~~2) turns out to be a "sanitary cordon" converging to zero in
probability, and redefining
a2(8, t) = .f.' .f.'(min(u, t» - ut')dH( u)dH( t», 0 < 8 < t < 1,
and
and introducing
a - { n 1/
2a{l/n, A'nln)n - n 1/ 2
, if a{l/n, knln) > 0,, otherwise,
•
()k I· /1..'"/11 (A' )1/2O<rn = ~ 1- n n sdH(s)< 1-2..
kn a{1nln, knln) . I" /11 n
and00
N{t) = L I(Sil) ~ f), 0 < t < 00,
k=l
the right-continuous version of the Poisson process ,,:,\T1 (.) in Sections 1 and 2, the proof
of the following main result (Theorems 1 and 2 in [11]) is obtained by an involved
combination of the techniques of [9] and [10]. i.e., those of the preceding two sections.
21
RESULT. (i) Assume tllat there exist a subsequence {n'} of the positive integers, a
left-continuous, non-decreasing function <p defined on (0,00) lrith <p(1) < 0 and .,,(1+) > e0, a left- continuous, non-decreasillg function ~! defined on (-00,00) ldtll ~'(O) < 0 and
11'{0+) > 0, and a constant 0 :5 6 < oc such that, as n' ~ 00,
(4.1) CPnl(S) ~ cp(s) at every continuity point S E (0,00) of <p,
(4.2) 1/'n' (x) ~ 'ljJ(x) at every cOlltinuity point x E R of 1/',
Then, necessarily, cp(s) < 6 for all S E (0, (0),
(4.4) .{"'(I'(S) -I'(",,)'ds < 00 for all c> 0,
and there exist a subsequence {n"} C {n'} and a sequence of positive nUlll-
bel's In" satisfying In" ~ 00 and In" /kn" ~ 0, as n" ~ 00, such that either ea(ln" /n", kn" /n") > 0 for all n", in Wllic11 case for some 0 < b < 6 and 0 < r <
/ 2 . c;; /I "(1_0)1 ,vn"a(ln"/n",kn,,/n")/AIl,, ~ b, rn" ~ r, ora(ln,,/n , kn,,/n ) = 0 for all
n", in 'which case 'we put b =r =0, and
{
k" }_1_ t X n"+1-j.n" - P/l" -.E.... V'(cp, t;" b, r, 0)An" j=1
as n" ~ 00, where a is zero or positive according to the hro cases, 'ljJ necessarily satisfies
(4.5)
and
tP(x) ~ -6r/(1 - 0), -00 < x < 00,
V(<p, 1/;, b, r,n) = rx. (.N(t) - t)dy(t) + t .N(t)d.p(t) + bZ1+ fO ~'(x)dx~.11 .10 .I-Z(r,o:)
22
where Z(r, 0) = -rZI +(1-0- r2)1/2 Z2, il"llere ZI and Z2 are standard normal random
variables SUdl that ZI, Z2 and N(·) are independent. l\!oreorer, if r.p =0 then b = 6,
while if 6 = 0 then r.p(s) = 0 for all s > l.
(ii) If there exist a subsequence {n'} of tIle positive integers and tlro sequences
AnI> 0 and Cnl along it such that
(4.6) 1 {k nl
}A I L "Ynl+l-j,nl - Cnln j=1
•
con"'erges in distribution to a non-degenerate limit, tllen there exists a subsequence
{n"} C {n'} such that conditions (4.1), (4.2) and (4.3) hold along the sequence {n"}
for An" in (4.6) and for approp!iate functions y and 1/' with the properties listed above
(4.1) and for some constant 0 < 6 < 00, with y satisfying (4.4) and t/; satisfying (4.5)
with an r E (0, (1 - 0)1/2) arising along a possible further subsequence. The limiting
random variable of the sequence in (4.6) is necessarily of the form V( y, 1/;, b, 7',0) +d for
appropriate constants 0 < b < 6, 0 < r < (1 - 0)1/2 and
d = lim (J.ln lll - enlll )/Anlll,n", ......oo
for some subsequence {n"'} C {n"}. l\Ioreorer, either r.p ¢ 0 or tt' ¢ 0 or b > O.
Just as in the case of full sums in Section 2, it is possible to see the effect on the
limiting distribution of deleting a finite number k > 0 of the largest summands from the
extreme sums En(kn), Replacing En(kn} by 'L/;;:-kn+l Xj,n, J.ln by
l kn/n (k+1)J.ln(k) = -n H(u)du - H -- ,
. (k+l)/n n
and the first two integrals in V(r.p, t/;, b, 7', a) by
00 5(1) k+lf (N(t) - t)dr.p(t) - f 1<+1 tdy(t) + kY(Sk~l) - f r.p(t)dt,.15 (1) • 1 . 1
1<+1 .
23
the result above remains true word for ,vord.
The result can be formulated for the sum of lower extremes 2:j~"l Xj,n, where
'Inn -+ 00 and 'lnn/n -+ 0 or 'Inn = [nl3] with 0 < f3 < 1. The limiting random variable
is of the form - V'( I{), 'ljJ, b, T, 13) with appropriate ingredients. In fact, if at least one of Q
and 13 is zero then the two convergence statements hold jointly with the limiting randOln
variables being independent.
One corollary of the result above is that the sequence in (4,6) converges in distribu
tion to a non-degenerate normal variable if ~nd only if (4.1) and (4.2) are satisfied v.'ith
An' =an', <p =0 and t/J =0, in ''v'hich case, choosing An' =an' and en' =Jln' in (4.6)
the limit is standard normal.
Another exhaustive corollary is the convergence in distribution of (En - Jln)/a n,
along the whole sequence {n}, when the underlying distribution is in the domain of one
of the three possible limiting extremal distributions in the sense of extreme value theory.
The details are contained in Corollary 2 in [11]. be}ng a common generalization of results
from [13], [15] and [40] mentioned in the introduction.
5. Related results
Here we mention some closely related developments obtained by the same proba
bilistic approach. References are given only if they are directly relevant to this approach,
further references can be found in the cited papers.
5.1 A generalization: L-statistics. !\·fason and Shorack [46,47,48] consider linear
combinations of order statistics of the form
n-k.. n-k..
2: Cjn~Yj,n 1) 2: CjnQ(Uj,n),j=m.. +l j=m ..+l
or more generallyn-k..
T~ = 2: Cjng(Uj,n),j=m.. +l
24
•
where g is some function and
[
j/n
Cjn = n J(t)dt, 1 < j < 71,. (i-l)/n
with some function J regularly varying at 0 and 1. In the moderate trimming case they
use a reduction principle sho'wing that the asymptotic distribution problem for T: is the
same as that forn-k..
Tn = L IqUj,n),j=m .. +l
where Ii, as a measure, is defined by dIi = Jdg. Thus, under certain conditions on g
and J, in [48] they obtain results parallel to those in [9] sketched in Section 3 above.
Even for fixed (light) trimming (or no trimming), in [46,47] they still obtain a theory
parallel to that in [10], sketched in Section 2 above.
5.2. Extreme and self-normalized sums in the domain of attraction of a
stable law. The paper [6] gives a unified theory of such sums based on the preliminary
e results in [4]. (A somewhat incomplete such theory was given earlier by LePage, \Yor
droofe and Zinn [39].) The .idea is that properly centered whole sums L:i=l Q(Uj) and
the individual extrems Q(U1,n), ... ,Q(Um,n),Q(Un-k,n,." ,Q(Un,n) converge jointly
with the same normalizing factor. This is trivial in our approach, where convergence is
in fact in probability. Paralleling a result of Hall [36], an approximation of an arbitrary
stable law by suitably centered sums being the asymptotic representations of the sum
of a finite number of extremes is given. (For a generalization, see the next subsection.)
The self-normalized sums are those considered by Logan, :Mallows, Rice and Shepp [..n]:
Ln(P) = CEi=l Xi)/(L:i=l IXiIP)l/p. The emphasis in [6] concerning this is the investi
gation of the properties of the limiting distribution using a representation arising out of
our approach. The extremes have a definite role in t~ese properties. In [16]', we used our. .
quantile approach to prove half of a conjecture in [41] stating that if F is in the domain
of attraction of a normal law and E..\ = 0, then Ln (2) ~ N(O, 1). This was proved
25
earlier by :Maller [42]. The converse half of the conjecture is still open. For further
results see [35]. e5.3. An "extreme-sum" approximation of infinitely divisible laws without
a normal component. Given an infinitely divisible distribution by its characteristic
function of the form of the right-side of (2.10) with l' replaced by a general constant
6, where L and R are left- and right-continuous Levy measures, respectively, so that
L(-oo) = 0 = R(oo) and
L: x2dL(x) +.f x'dR(x) < 00 for any E > 0,
we see that forming 'P1(5) = inf{x < 0: L(x) > 5}, 0 < 5 < 00 and <;'2(5) = inf{x < 0:
-R{-x) > 5}, 0 < 5 < 00, so that (2.2) is satisfied, the random variable lo,o + e- 'Y
has the given infinitely divisible distribution. From now on suppose that p = O~ and
let Fo{') = FO{'P1, 'P2, 6;·) be the distribution f~nction of lo,o -+ 6 - 'Y. The approach
sketched in Section 2 implies that under the said conditions
converges in distribution along some subsequence to
So the second component here represents the asymptotic contribution of the extrems in
the limiting infinitely divisible law of the full sum. Hence it is conceivable that a suitably
centered form of this second component can approximate now Vo,o if m, k -+ 00.
Let Lm,k be the Le'vy distance between Fo and the distribution function of
26
..
..
Then it is shown in [7] that Lm,k --+ 0 as nI, It' --+ 00, and, depending on how fast ~l (s)
and 'P2(S) converge up to zero as S --+ 00, rates of this convergence are also provided
which rates are sometimes amazingly fast. In the special case of a stable distribution
with exponent 0 < Q' < 2, given by 'Pj{s) = -CjS-I/o, S > 0, Cl,C2 ~ 0, Cl + C2 > 0,
when Lm,k can be replaced by the supremum distance ](m,k, we obtain
where °< e < 1 is as close to 1 as ,ve wish,
5.4. Almost sure behaviour: stability and the law of the iterated loga
rithm. Since this topic is so broad that it could be reviewed only in a separate sun-ey,
here we restrict ourselves to mention that Haeusler and l\Iason [29,30,31], Einmahl,
Haeusler and l\fason [24] and Haeusler [28] use the quantile transform - empirical pro
cess method to investigate the almost sure behaviour of sums considered in [13] and [15],
and when the underlying distribution has a slowly varying tail. Here the basic approach
should be combined with techniques which go back to Kiefer [37] and Csaki [2] and some
further strong developments such as Deheuvels [20]. The same method is used to prove
a universalliminf law by ~Iason [45].
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