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].

References

[1] D.Z. AROVand A.A. BOBROV, The extreme terms of a sample and their role in

the sum of independent variables. Theory Probab. Appl. 5(1960), 377-369.

[2] E. CS..\KI, The law of the iterated logarithm for normalized empirical distribution

functions. Z. Wahrsch. Verw. Gebiete 38(1977), 147-167.

[3] M. CSORGO, S. CSORGO, L. HORVATH and D.M. l\lASON, \Yeighted empirical

and quantile processes. Ann. Probab. 14(1986), 31-85,

27

[4J.M. CSORGO, S. CSORGO, L. HORVATH and D.:M. 1LASON, Normal and stable

convergence of integral functions of the empirical distribution fUllction. Ann. Probab.

14(1986), 86-118.

[5] M. CSORGO, S. CSORGO, L. HORVATH and DJvI. 1IASON, Sup-norm COllver­

gence of the empirical process indexed by functions and applications. P.robab. Math.

Statist. 7(1986), 13-26.

[6] S. CSORGO, Notes on extreme and self-normalised sums from the domain of attrac­

tion of a stable law. J. London Math. Soc. (2), 39(1989), 369-384.

[7] S. CSORGO, An extreme-sum approximation to infinitely divisible laws without a

normal component. In: Probability on Vector Spaces IV (S. Cambanis and A. \Yeron,

eds.) Lecture Notes in Afathematics. Springer, Berlin. ro appear.

[8] S. CSORGO, P. DEHEUVELS and D.:M. ~IASON, Kernel estimates of the tail index

of a distribution. Ann. Statist. 13(1985), 1050-1077. e[9] S. CSORGO, E. HAEUSLER and D.~f. 1\IASON, The asymptotic distribution of

trimmed sums. Ann. Probab. 16(1988), 672-699.

[10] S. CSORGO, E. HAEUSLER and D.M. :MASON, A probabilistic approach to the

asymptotic distribution of sums of independent, identically distributed random vari­

ables. Adv. in Appl. Math. 9(1988), 259-333.

[11] S. CSORGO, E. HAEUSLER and D.1\!f. 1IASON, The asymptotic distribution of

extreme sums. Ann. Probab. 18(1990), to appear.

[12] S. CSORGO, E. HAEUSLER and D.1\I. :MASON, The quantile-transform approach

to the asymptotic normality of modulus trimmed sums. Preprint.

28

[13] S. CSORGO, L. HORVATH and D.~I. !\IASON, \Vhat portion of the sample makes

a partial sum asymptotically stable or normal? Probab. Theory ReI. Fields 72(1986),

1-16.

[14] S. CSORGO and D.!\'L 1lASON, Central limit theorems for sums of ext;eme values.

Math. Proc. Cambridge Philos. Soc. 98(1985), 547-558.

[15] S. CSORGO and D.M. 1tIASON, The asymptotic distribution of sums of extreme

values from a regularly varying distribution. Ann. Probab. 14(1986), 974-983.

[16] S. CSORGO and D.:M. ~vIASON, Approximations of weighted empirical processes

with applications to extreme, trimmed and self-normalized sums. In: Proc. First

World Congress Bernoulli Soc. Vol. 2, pp. 811-819. ,VNU Sci. Press, Ctrecht, 1987.

[17] S. CSORGO and D.M.11ASON, Simple estimators of the endpoint of a distribution.

In: Extreme Value Theory, Oberwolfach 1987 (J. Hiisler and R.-D. Reiss, eds.) pp.

e 132-147. Lecture Notes in Statistics 51. Springer, Berlin.

[18] S. CSORGO and D.~1. 11ASON, Bootstrapping empirical functions. Ann. Statist.

17(1989), No.4, to appear.

[19] D.A. DARLING, The influence of the maximum term in the addition of independent

random variables. Trans. Amer. Math. Soc. 73(1952), 95-107.

[20] P. DEHEUVELS, Strong laws for the k-th order statistic when k < clog2 n. Probab.

Theory ReI. Fields 72(1986). 133-156.

[21] V.A. EGOROV, The central limit theorem in the absence of extremal absolute order

statistics. (In Russian) Zap. Na·uchn. Sem. Leningrad Otdel. Mat. Inst. Steklo'Y

(LOMI) 142(1985), 59-67.

29

[22) V.A. EGOROV and V.B. NEVZOROV, Some rates of con\"ergence of sums of order

statistics to the normal law. (In Russian) Zap. Nauchn. Sem. Leningrad Otdel. Mat. eInst. Steklov (LOM!) 41(1974), 105-128.

[23) V.A. EGOROV and V.B. NEVZOROV, Sununation of order statistics and the

normal law. (In Russian) Vestnik Leningrad Univ. Mat. Mekh. Astronom. 1974 No.

1, 5-11.

[24) J.H.J. EIN11AHL, E. HAEUSLER and D.~L ~lASON,On the relationship between

the almost sure stability of weighted empirical distributions and sums of order

statistics. Probab. Theory Rei. Fields 79(1988), 59-74.

[25) B.V. GNEDENKO and A.N. KOL:MOGOROV, Limit Distributions for Sums of

Independent Random Variables. Addison-'Yesley, Reading, rvlass., 1954.

[26) P.S. GRIFFIN and 'V.E. PRUITT, The cel;ltrallimit problem for trimmed sums.

Math. Proc. Cambridge Phi/os. Soc. 102(1987), 329-349.

[27) P.S. GRIFFIN and 'V.E. PRUITT, Asymptotic normality and subsequentialliinits

of trimmed sums. Preprint.

[28) E. HAEUSLER, Laws of the iterated logarithm for sums of order statistics from a

distribution with a regularly varying upper tail. Habilitationsschrift, University of

Munich, 1988.

[29) E. HAEUSLER and D.rvI. MASON, Laws of the iterated logarithm for sums of

the middle portion of the sample. Math. Proc. Cambridge. Philos. Soc. 101(1987),

301-312.

[30) E. HAEUSLER and D.M. l\fASON, A law of the iterated logarithm for sums of

extreme values from a distribution with a regularly varying. upper tail. Ann. Probab.

15(1987), 932-953.

30

•

•

[31] E. HAEUSLER and D.11. :MASON, On the asymptotic behavior of sums of order

statistics from a distribution with a slowly varying upper tail. Preprint.

[32] :M.G. HAHN and J. KUELBS, Universal asymptotic normality for conditionally

trimmed sums. Statist. P.robab. Letters. To appear.

[33] :M.G. HAHN, J. KUELBS and J.D. SA!vIUR, Asymptotic normality of trilluned

sums of ~-mixing random variables. Ann. Probab. 15(1987). 1395-1418.

[34] M.G. HAHN, J. KUELBS and D.C. \VEINER, On the asymptotic distribution of

trimmed and winsorized sums. Preprint.

[35] M.G. HAHN, J. KUELBS and D.C. \VEINER, Self-normalization of censored

sums and sums-of-squares in joint estimation of certain center and scale sequences.

Preprint.

[36] P. HALL, On the extreme terms of a sample from the domain of attraction of a

stable law. J. London Math. Soc. (2), 18(1978), 181-191.

[37] J. KIEFER, Iterated logarithm analogues for sample quantiles when Pn ! o. Proc.

Sixth Berkeley Symp. Vol. 1, 227-244.

[38] J. KUELBS and M. LEDOUX, Extreme values for vector valued random variables

and a Gaussian central limit theorem. Probab. Theory Rei. Fields 14(1987),341-355.

[39] R. LEPAGE, IvL WOODROOFE and J. ZINN, Convergence to a stable distribution

via order statistics. Ann. Probab. 9(1981), 624-632.

[40] G.S. LO, A note on the asymptotic normality of sums of extreme values. J. Statist.

Planning Inference 22(1989), 127-136.

31

[41] B.F. LOGAN, C.L. 1fALLO\VS, S.O. RICE and L.A. SHEPP, Limit distributions

of self-normalized sums. Ann. Probab. 1(1973), 788-809.

[42] R.A. MALLER, A theorem on products of random variables, with application to

regression. Austral. J. Statist. 23(1981),177-185.

[43] R.A. MALLER, Asymptotic normality of lightly trimmed means - a con'-erse. Math.

Proc. Cambridge Philos. Soc. 92(1982), 535-545.

[44] R.A. MALLER, Asymptotic normality of trimmed means in higher dimensions. Ann.

Probab. To appear.

[45] n.1f. 11ASON, A universal one-sided law of the iterated logarithm. Preprint.

[46] n.1'I. ~IASON and G.R. SHORACK, Necessary and sufficient conditions for asymp- ,

totic normality of L-statistics. Preprint.

[47] n.M. 1.1ASON and G.R. SHORACK, Non-normality of a class of random variables. ePreprint.

[48] n.M. :MASON and G.R. SHORACK, Necessary and sufficient conditions for the

asymptotic normality of trimmed L-statistics. Preprint.

[49] T. MaRl, On the limit distributions of lightly trimmed sums. Math. Proc. Cambridge

Philos. Soc. 96(1984). 507-516.

[50] W.E. PRUITT, The class of limit laws for stochastically compact normed sums.

Ann. Probab. 11(1983), 962-969.

[51] \V.E. PRUITT, Sums of independent random variables with the extreme terms

excluded. Preprint.

32

..

[52} H.-J. ROSSBERG, Uber das asymptotische Verhalten der Rand - und Zentralglieder

,e einer Variationsreihe II. Publ. Math. Debrecen 14(1967), 83-90.

[53] G.R. SHORACK and J.A. '\lELLNER, Empirical Processes with Applications to

Statistics. Wiley, New York, 1986.

[54] S.M. STIGLER, The asymptotic distribution of the trimmed mean. Ann. Statist.

1(1973), 472-477.

[55] J.L. TEUGELS, Limit theorems on order statistics. Ann. Probab. 9(1981), 868-880.

[56] V.V. VINOGRADOVand V.V. GODOVAN'CHUK, Asymptotics oflarge deviations

for sums of independent random variables after deleting the extreme summands. (In

Russian) Teor. Verojatn. Primenen. To appear.

Bolyai Institute

University of Szeged

Aradi vertanuk tere 1

H-6720 Szeged, Hungary

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