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Journal of Statistical Planning and Inference 28 (1991) 305-317 North-Holland 305 Asymptotic behaviour of linear combinations of functions of order statistics Rimas NorvaiSa Department of Mathematics, University of Vilnius, Naugarduko 24, Vilnius 6, Lithuania RiEardas Zitikis Institute of Mathematics and Cybernetics, Lithuanian Academy of Science, Akademijos 4, Lithuania Received 2 August 1989; revised manuscript received 20 August 1990 Recommended by M.L. Puri Vilnius 600, Abstract: Sufficient conditions for an L-statistic to be decomposed into the sum of independent real random variables and the remainder term with the asymptotically optimal upper bound are given. This result leads to less stringent conditions for some limit theorems. AMS Subject Classification: Primary 62G30; secondary 60B12. Key words and phrases: L-statistic; law of the iterated logarithm; Banach space valued random variables. 1. Introduction Let X,X,, X2, . . . be independent identically distributed (i.i.d.) random variables (r.v.‘s) with a common distribution function (d.f.) Fand (Q, @,P) be the underlying probability space. In this paper we deal with an asymptotic behaviour of the L-statistic, i.e. the linear combination of the function of order statistics Tn = t ,c, cinh(xi:n)7 nerd, , (1.1) where X, :n denotes the i-th order statistic of X,, . . . ,X,, h is a real measurable func- tion and cl,,, . . ..c., are certain real coefficients. For given weights generating function J we denote by T,” the L-statistic (1.1) with coefficients I ) i/n 0 Gin = n J(u)du, i= l,..., n. s (r- 1)/n 0378-3758/91/$03.50 % 1991-Elsevier Science Publishers B.V. (North-Holland)

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Page 1: Asymptotic behaviour of linear combinations of functions of order statistics

Journal of Statistical Planning and Inference 28 (1991) 305-317

North-Holland

305

Asymptotic behaviour of linear combinations of functions of order statistics

Rimas NorvaiSa

Department of Mathematics, University of Vilnius, Naugarduko 24, Vilnius 6, Lithuania

RiEardas Zitikis

Institute of Mathematics and Cybernetics, Lithuanian Academy of Science, Akademijos 4, Lithuania

Received 2 August 1989; revised manuscript received 20 August 1990

Recommended by M.L. Puri

Vilnius 600,

Abstract: Sufficient conditions for an L-statistic to be decomposed into the sum of independent real

random variables and the remainder term with the asymptotically optimal upper bound are given. This

result leads to less stringent conditions for some limit theorems.

AMS Subject Classification: Primary 62G30; secondary 60B12.

Key words and phrases: L-statistic; law of the iterated logarithm; Banach space valued random

variables.

1. Introduction

Let X,X,, X2, . . . be independent identically distributed (i.i.d.) random variables

(r.v.‘s) with a common distribution function (d.f.) Fand (Q, @,P) be the underlying

probability space. In this paper we deal with an asymptotic behaviour of the

L-statistic, i.e. the linear combination of the function of order statistics

Tn = t ,c, cinh(xi:n)7 nerd, ,

(1.1)

where X, :n denotes the i-th order statistic of X,, . . . ,X,, h is a real measurable func-

tion and cl,,, . . ..c., are certain real coefficients.

For given weights generating function J we denote by T,” the L-statistic (1.1)

with coefficients

I

) i/n 0

Gin = n J(u)du, i= l,..., n. s (r- 1)/n

0378-3758/91/$03.50 % 1991-Elsevier Science Publishers B.V. (North-Holland)

Page 2: Asymptotic behaviour of linear combinations of functions of order statistics

306 R. NorvaiSa, R. Zitikis / Linear combinations of functions of order statistics

In passing, this statistical function, as well as more general one (see (2.1) below),

is best suited to handle under our technique. Another common subclass of (1.1) con-

stitute the L-statistics denoted by

T$i/ -$ (‘1 h(X,:,), n e IN. n,=l .

(1 .a

This type of statistic is treated here by shifting to T,“.

The usual approach to the asymptotic analysis of the statistics just introduced in-

volve the decomposition in a leading and remainder terms. Namely, any statistic T, we may put down in the form

(1.3)

where p is a constant and Z,, i= 1, . . . , n are i.i.d. r.v.‘s with mean zero and finite

variance. Assume R, = O,(d,,) to hold for some sequence d, 10. Then asymptotic

normality of fi(T,-p) follows from the classical central limit theorem (CLT) if

fid, 4 0, as n+ + 03 (see e.g. Chernoff, Gastwirth and Johns (1967), Stigler

(1974), Shorack (1972), Govindarajulu and Mason (1983)). Analogously other limit

theorems can be derived if d, 10 sufficiently fast. Therefore it is interesting to get

an optimal bound of R, under less stringent conditions as possible (see Ghosh

(1972)).

Let us restrict our attention for a moment to the L-statistic (1.2) with h(x) =x.

It was shown in Ghosh (1972) assuming

(i) J” is bounded,

(ii) EIXIP<+w for somep>2,

that

d, = log2 n/n, n E N.

This has been improved in Singh (1981) showing that

d, = log: n/n, n E iN, (1.4)

(here and throughout the article we write log+ x to denote log(xv e) and use log: x

to denote log+ (log+ x)) if

(i) J” is bounded except possible at finitely many points,

(ii) E IXIp< + 03 for some p> 1 and some assumptions on F in a neighborhood

of each of the excluded point are imposed.

Moreover, if the condition E jX j < + 03 is satisfied instead of (ii) then

d, = (logn)‘+&/n, nEN, Ve>O. (1.5)

Another improvement was given in Lea and Puri (1988). Namely, it was proved that

(1.5) holds if

(i) J’ exists and satisfies a Lipschitz condition,

(ii) EIXI< +m. (1.6)

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R. NorvaiSa, R. Zitikis / Linear combinations of functions of order statistics 307

In this paper the general L-statistic (1.1) is treated. Particularly it follows from

Corollary 2.8 that for the L-statistic (1.2) with h(x) =x we have (1.4) if

(i) J satisfies a Lipschitz condition,

(ii) for all f E L,,

a2(f) := I\ f(s)f(r)[F(s)AF(I)-F(s)F(t)] dsdr< + 03, I I

EIXl/log,f jXI< +a.

Note that both conditions in (ii) are strictly weaker than (1.6). We show also that

such asymptotic behaviour can’t be improved for the L-statistic 7”‘. An approach

involves only arguments, which are essentially elementary, and is based on the law

of the iterated logarithm (LIL) in Banach spaces. It was Bentkus and Zitikis (1990)

have used firstly this approach. Usually results for U-statistics or uniform empirical

processes are explored (see e.g. Helmers (1982), Serfling (1980), Vandemaele and

Veraverbeke (1982)).

We conclude this section by fixing some notation. For real functionfon [0, I] we

write f l H(a), a E [0, I], if there exists a constant M< + ~0 such that

1f(U)-f(U)lIM. Iu-UI” Vu,oE[O,l],

i.e. if f satisfies H(ilder’s condition of order cr. The indicator function of a set A

is denoted by 1,4. F,, denotes the empirical d.f. based on the sample X1, . . . ,X,, i.e.

F,(t) = .-I i 1 ;=, Io,tl(x;).

2. Main results

The basic assumption underlying our results which will be assumed to hold

without reference is the following one: the measurable function h is non-decreasing,

defined on (a,b) with a=inf(t:F(t)>O}, b=sup{t:F(t)<l}, and such that the

a-finite Lebesgue-Stieltjes measure mh on (a, b) generated by h is properly defined.

This remark leads us to investigate the separable Banach space L,(h) of those real

measurable functions f on ((a, b), mh) for which 1 f 1 p is integrable. We use the

following convention, concerning the integral sign:

i I ’ f dh = ‘$-(t) dh(t),

<a

whenever this integral make sense as Riemann-Stieltjes integral.

As mentioned in the introduction we consider the general statistical functional

defined by

I,,(v) = [woF,- WOFI dh, (2.1)

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308 R. NorvaiSa, R. Zitikis / Linear combinations of functions of order statistics

where I,V is some real measurable function on IO, 11. We assume in the sequel that

Z,(w) exists and is finite. The next lemma formulates a fairly obvious fact that if

the integral

,LP= - hdyOF 1

(2.2)

exists then I,(w) is properly defined too.

Lemma 2.1. Let w be monotone function. Moreover, let h or ly and F be con- tinuous. Suppose that

and

lim h(t)[ly(l) - v(F(t))l = 0 r-b

lim h(t)[ty(O) - ty(F(t))] = 0. ,+a

If the integral ,uV exists then I,(y) exists too and

i-l In(w) = -pY+;g* y ~ H > n

(2.3)

The proof of the lemma is actually just the usual proof of the formula of partial

integration. All integrals from A to B, with A and B being any real numbers such

that a < A <X, : ,, and X,: n <B < b, exists on account of the assumptions on h, v/, and F. Letting A + a and B-+ b yields the statement of Lemma 2.1.

Particularly, if the weights generating function J is integrable and

‘I

we(x) = \

J(u) du, x~lO,ll, (2.4) LX

then the sum in the right-hand side of (2.3) with w. instead of I+U is L-statistic T,“.

The equality (2.3) in this case was stated in Govindarajulu and Mason (1983) under

the additional assumption that the integral Z,(v) is properly defined.

Now we decompose In(w) into a leading term with a simple structure and re-

mainder term which is majorized by the Cramer-von Mises type statistic. For each

n E k. and any function w define the r.v.‘s

Zw = [l(,,,,(X;) - F(t)] ty’(F(t)) dh(t), i = 1, . . . , n. (2.5)

Lemma 2.2. Assume the function y to be differentiable with the derivative I,V’E H(a) for some a E [0, I]. Then

I,(W) =’ i Zy+r,, n ;=I

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R. NorvaiSa, R. Zitikis / Linear combinations of functions of order statistics 309

where I,,(w) and .Zlw are given by (2.1) and (2.5), respectively. Moreover, for all

r?EtN,

Irnl <MS \ IF,-Fl”“dh. (2.6)

Proof. By the Mean Value Theorem, we have the following decomposition:

Z,(w)= \ @o(BF,+(l-B)F)[F,-F]dh

for some I!?E (0,l). Using Holder’s condition we get (2.6) and this completes the

proof.

The domination (2.6) suggests that the LIL for L, +.(h)-valued r.v.‘s can be used

to handle the asymptotic behaviour of the remainder term. Let us recall now some

results regarding the LIL behaviour in the Banach space setting.

Lemma 2.3 (Goodman, Kuelbs and Zinn (1981)). Let (H, 11 . 11) be the separable Hilbert space and let y, y,, y,, . . . be i.i.d. H-valued r.v. ‘s. Then y satisfies the

bounded LIL, i.e.

(2.7)

iffEf(y)=O, Ef2(y)< +03, for alffeH, andE{llyl12Aog: I/yll}< +a.

Now we quote a sufficient condition for the LIL to hold in the non-Hilbert space

L I+.(h), a~[O,l).

Lemma 2.4. Let p E [l, 2) and y be a mean zero L,(h)-valued r.v. Then y satisfies the bounded LIL (2.7) if

0

I (~Fy~(t))~‘* dh(t) < + ~0. (2.8) .,

Remark. Condition (2.8) means that a centered L,(h)-valued r.v. y is pre-Gaussian,

i.e. its covariance is that of a tight Gaussian probability measure on L,(h) (see the

last section for the supplement information).

Proof of Lemma 2.4. Follows from Theorem 4.3 in Pisier (1975) (or Heinkel

Page 6: Asymptotic behaviour of linear combinations of functions of order statistics

310 R. NorvaiSa, R. Zifikis / Linear combinations of functions of order statistics

(1979), Goodman, Kuelbs and Zinn (1981)) and the CLT in cotype 2 Banach spaces

(see e.g. Araujo and GinC (1980)).

To state the following auxiliary result, we use the class of real r.v.‘s which con-

stitutes the Lorentz space L,,, O<p,q< +tcx, i.e. the class of all real r.v.‘s such

that

‘m 114

IIqJ,, := (I r@(P({ IX1>r}))4"'dt >

<t-m. LO

(2.9)

Note that L,,, is just L, by the usual integration by parts formula and L,,, C L,,,, ,

if q < q’. Often in statistics conditions similar to

i x*~‘[F(x)(l -F’(~))]~“‘dx< + 03 (2.10)

are used. It is easy to see that this condition and (2.9) are equivalent. In other words,

(2.10) means that XEL~,~. Ordinary the Lorentz spaces help to compare the tails

and supplement the information given in the remark on p. 276 of Serfling (1980).

Lemma 2.5. Suppose h(X)EL,,,,, for some O<p< + 03. Then there exist con- stants c, >O and c2 such that

CI Ih(X)l”‘~ IIYII L,,(/7)s lh(X)l”“+c2 a.s.,

where

~(0 := lc,,,l(X) -F(t), fe (4 b). (2.11)

Proof. Let us start by establishing the second inequality. We may and do assume

that h(t,) = 0 for some to E (a, 6). Put

M := [F@,)]_ v [F(r,)]_‘.

where F(t) = 1 -F(t). Then for any t~(a, b) we have

I _w>l = 1(,,X]wYo+ l[X,b)wm

I M. F(t)F(f) + l,,&), if X5 to,

l~t,,~j@), if Xrh.

This gives rise to the inequality

I/ Yll L,(+M. IIF.%p(h)+ lW)l”p.

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R. NorvaiSa, R. Zitikis / Linear combinarions of functions of order statistics 311

Let us consider now the lower estimation of IIY~/&). For t, as above, we have

II Yll &/l) 2 4a,r”)w) ’ 1 .\ ,x ,,,,, F” dh + h,,,,(X) lI/,,,x]FP dh

2 bTfo)~F(t")lP Ih(X

This completes the proof of lemma.

Now we are ready to state and prove the main result.

Theorem 2.6. Assume the function I,U to be differentiable with the derivative V’E H(a) for some a E [0, I] and the constants cln, . . . , c,, to be such that

rn~~cjR-n[~(+)-.(~))/ =O(nP), asn+ fw. (2.12)

Let X,X1,X1, . . . be i.i.d. r.v. ‘s with the d.f. F such that (i) for all f E L_,(h),

o,“(f) := II

f(s)f(t)[F(s)/\F(t)-F(s)F(t)] dh(s)dh(t)< +w, I ,)

E lh(X)I/log; Ih( < + 03,

in the case a = 1, and

(ii) h(X)EL2/Cl+n),~9 i.e.

I ‘m(P({lh(X)l>t}))(1+U)‘2dt< +03,

CO

(2.13)

in the case a~ [0, 1).

Then for the L-statistic (1.1) we have the representation

where uuw and Zp are given by (2.2) and (2.5), respectively, and the remainder term is such that

R, = :I O,(n-’ log: n),

Q&t-“),

Moreover, for the statistic defined

if o= 1,

if aE[O,l).

by

(2.14)

T,w := j, [ .(?)-v(;)]W;:,J we have the representation

Page 8: Asymptotic behaviour of linear combinations of functions of order statistics

312 R. Norvaiga, R. Zitikis / Linear combinaiions of functions of order statistics

with the remainder term

r, = Op([log,+n/n]” +a)‘2).

Proof. By virtue of Lemma 2.1 and Lemma 2.2, we can write

=; ,$, Zy+R,, I

with the remainder term R, such that

lRnl SM. I~~-~~‘+adh+; .r

=: Z,(n)+Z2(n).

First, consider Z,(n). We rewrite Z,(n)

I;, ~in-n~~(~)-W(i)])h(X’:“)l (2.15)

into the following form:

where, as in (2.11),

y;(t):=l~,,,,(x;)-F(t), i= l,..., n, tE(a,b).

By the bounded LIL, we have

lim sup (n/log2f n) (‘+a)‘2Z1(n)< + 05 a.s. n+orr

(2.16)

if appropriate conditions for the L, +a (h)-valued r.v. y are satisfied. Using Lemma

2.6 and the equivalence of (2.8) to

I ’ F(l -01 (1+a)‘2dh< +m,

it is easy to check that conditions of Lemma 2.3 (case (Y= 1) and Lemma 2.4 (case

a E [0, 1)) are satisfied due to the assumptions (i) and (ii), respectively. Second, we

consider Z2(n). By (2.12), we have

Z,@Z)<C~+~+~) i Ih(_X;)l. r=l

In the case o E [0, 1) the inclusion L 2,o + a), 1 c L, holds. Consequently E 1 h(X) I< + CO and, by the strong law of large numbers, Z2(n) = Or(nP). This in conjunction with

(2.16) gives the bound for (2.15) thereby proving (2.14) if a E [0, 1). Assume now

a= 1. It is sufficient to show

,‘~IJII hnns~p P (i ,c, Ih( >A. a@)]) = 0, (2.17)

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R. Norvaiia, R. Zitikis / Linear combinations of functions of order statistics 313

where a(t) = t + log: t, t E R+. Note that the inverse function a-‘(t) is asymptotically

equivalent, as t+ + 03, to the function t/log: 1. Hence, by the assumption (2.13),

we have

lim sup n. P({ Ih(X)I >a(n)}) = 0. n-c=

This implies

lim+yp P (1

$, Ih( >A. a(n)])

5 lim sup@ log:W’E IWW Lt0,,,,j1(lW)I) n-m

5 C. A-’ . E Ih(X)l/log; IA(

for some finite constant C. Hence we have (2.17). Moreover, taking into the account

(2.16) we see that (2.14) holds if (Y = 1 too. The proof of the last statement for the

statistic T,W goes the lines of the proof given above with Iz(n) = 0. This completes

the proof of Theorem 2.6.

Remark. The proof of Theorem 2.6 is particularly simple if the measure mh is

bounded on (a, 6). This is a case when the r.v. X is uniformly distributed on [0, l]

and h(x) =x, for example. Then Kiefer’s LIL for the supremum of the absolute

value of the empirical process yields at once

lrnj SC. sup lF,-FJ’+U = O,([log:n/n](l+““2).

Next we state the corollary for the function IJ given by (2.4) since we get the

L-statistic TJ in that case.

Corollary 2.7. Let the weights generating function JE H(cr), for some (Y E [0, 11, and

x,x,,x,, . . . be i.i.d. r.v.‘s with the d.f. F as in Theorem 2.6. Then the L-statistic

T,” := ,i, 1”’ J(u) du h(X;:,) I (lb 1)/n

has the representation (1.3) with

,V = JoF. hdF, Zj= - (lIo,.,(Xj)-F)JoFdh, i= l,..., n, (2.18) I i I’ ,I

and the remainder term

R, = O,([log: n/n]” +cO’2).

Now we will show that the bound (2.19) can’t be

I/ ,, . . . , U,, be a sample from the uniform distribution on

-2u, for u E [0, 11. Using above representation we have

(2.19)

improved for a= 1. Let

[O,l], h(x)=x and J(u)=

Page 10: Asymptotic behaviour of linear combinations of functions of order statistics

314 R. NorvaiSa, R. Zitikis / Linear combinations of functions of order statistics

R,= T,p-p-i i z, n ;=I

” = i

([F;(u) - u2] - 2[F,(u) - u]u) du <O

=

I

" [F,,(u)-U]2du. %O

Note that nR, is the Cramer-von Mises statistic. Thus

lim sup (n/log: n)R, = 2/7r2 a.s. n-m

(see Finkelstein (197 1)).

Let us apply Theorem 2.6 to the L-statistic TJ given by (1.2). In the case (x= 1

the condition (2.12) for the function t,u given by (2.4) means

And it is satisfied if the weights generating function J is the Lipschitz function on

(a,6). So, we get:

Corollary 2.8. Let the weights generating function J be the Lipschitz function and

xx,,x,, . . . be i.i.d. r.v. ‘s with the d.f. F as in Theorem 2.6. Then the L-statistic (1.2) has the representation (1.3) with u and Z;, i = 1, . . . , n, given by (2.18) and the remainder term

R, = O,( log: n/n).

Following Lea and Puri (1988) we apply our results to derive the LIL and the CLT

for the L-statistic T,‘. We consider T,’ instead of the more general statistic (1.1) for

the sake of simplicity only. Analogous statement for the L-statistic T, can be stated

also in an obvious manner. Thus applying Corollary 2.8 and the classical LIL we

obtain:

Corollary 2.9. Under the assumptions of Corollary 2.8 assume in addition that O<a2:=ai(hoF)< +a. Then

lim sup (2a2 log2n))“2fi IT,’ -,u 1 = 1 a.s. n+m

Another consequence of Corollary 2.8 and the classical CLT is the following

asymptotic normality of Td.

Corollary 2.10. Under the assumptions of Corollary 2.8 assume in addition that O<a’:=ai(hoF)< +a~. Then

fi(Td -u) 5 N(0,a2) as n - +cr>.

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R. NorvaiSa, R. Zitikis / Linear combinations of functions of order statistics 315

3. Concluding remarks

As noted in the introduction the key point in our approach is the use of the

bounded LIL in the Banach space setting. In addition to results stated above there

are also necessary and sufficient conditions for the LIL in any Banach space B to

hold (see Ledoux and Talagrand (1986)). According to this result B-valued r.v. y

satisfies the bounded LIL (i.e. (2.7)) iff Ef’(v)< + ~0 VfE B*, El~y~l’/log~ llyll<

+ 03 and the sequence

[~~$r_y,~~/~/%%&,n~N~ is bounded in probability. (3.1)

To apply this result, we need to verify condition (3.1) in the Banach space L,, 11p<2. In general form it seems to be difficult task. Therefore we use a stronger

condition which ensure also the CLT to hold.

However, as a byproduct, we find a simple condition for the process y given by

(2.11) to have a strong second order and to be pre-Gaussian in L,(h), pr 1. So, as

follows from Lemma 2.5 and the proof of Theorem 2.6, we have

E II Yllt,,(h) < fcx, iff ~(X)EL,,~,

(Ey2(t))“” dh(t)< + 03 iff h(X) EL,,,, , . (3.2)

This illustrates the difference between the pre-Gaussians condition and the strong

second order condition. Let us recall that the pre-Gaussians condition implies the

finiteness of the second strong order in the cotype 2 Banach spaces, which are

L,(h), 1 rp5 2, and vice versa in the type 2 Banach spaces, which are L,(h),

25p<cQ.

In Pisier and Zinn (1978) it is proved that mean zero L,-valued, p>2, r.v. y

satisfies the CLT iff in addition to (3.2) the following condition holds:

t2P({/1yll,,,>t})+0 as t-t +co. (3.3)

By Lemma 2.5, (3.3) is equivalent to

t2’pP({ Ih(X)l>t})-0 as f~ +OJ.

But the last condition for h(X) is weaker than (3.2). In the case 11~12, it is known

(see e.g. Araujo and GinC (1980)) that only (3.2) is sufficient (and necessary) for the

&-valued r.v. y to satisfy the CLT. Hence we have proved the following

statement.

Theorem 3.1. Let 1 sp< + 03 and G, be the empirical d.f. based on a sample

u, u,, . ..9 U,, from the uniform distribution on [O,l]. Then

~({~(G,(t)-t)},.co,1)) 5 WB(t)l,,~,,,,) as n ++w,

Page 12: Asymptotic behaviour of linear combinations of functions of order statistics

316 R. NorvaiSa, R. Zitikis / Linear combinations of functions of order statistics

on L,(h) iff h(U) E L2,,, 1, where B is the Brownian bridge.

As an easy consequence we get the statement related to the results from Cs6rg8

and Horvath (1988). Let the function q: (0,l) + IR+ be non-decreasing on (O,+],

q(t) = q(l -t), for t E [$, l), and be integrable on [E, 1 -E] V&>O.

Corollary 3.2. Let 1 sp< + 03 and q be as above. If

I [t(l - t)] p’2/q(t) dt < + co

then ’

I ‘I

Ifi(G,(t)-t)l”/q(t)dt 5 / ” IB(t)lP/q(t) dt as n - + 03.

*O I, 0

Proof. Take h(t) =jl,* (l/q(s)) d s and use Theorem 3.1 for the continuous func-

tional on L,,(h) given by the norm. This completes the proof.

References

Araujo, A. and E. Gine (1980). The Central Limit Theorem for Real and Banach Valued Random

Variables. John Wiley, New York.

Bentkus, V. and R. Zitikis (1990). Probabilities of large deviations for L-statistics. Liet. Mat. Rink. 30,

479-488.

Chernoff, H., J.L. Gastwirth and M.V. Johns, Jr. (1967). Asymptotic distribution of linear combina-

tions of order statistics with applications to estimation. Ann. Math. Statist. 38, 52-72.

Cskg’6, M. and L. Horvath (1988). On the distributions of L,, norms of weighted uniform empirical and

quantile processes. Ann. Probab. 16, 142-161.

Finkelstein, H. (1971). The law of the iterated logarithm for empirical distributions. Ann. Maih. Statist.

42, 607-615.

Ghosh, M. (1972). On the representation of linear functions of order statistics. Sankhya Ser. A 34,

349-356.

Goodman, V., J. Kuelbs and J. Zinn (1981). Some results on the LIL in Banach space with applications

to weighted empirical processes. Ann. Probab. 9, 713-752.

Govindarajulu, 2. and D.M. Mason (1983). A strong representations for linear combinations of order

statistics with applications to fixed-width confidence intervals for location and scale parameters.

Stand. J. Statist. 10, 97-l 15.

Heinkel, B. (1979). Relation entre theoreme central-limite et loi du logarithme it&e dans les espaces de

Banach. Z. Wahrsch. Vet-w. Gebiete 49, 211-220.

Helmets, R. (1982). Edgeworth Expansions for Linear Combinations of Order Statistics. Math. Cenire

Tracts No. 105. Mathematical Centre, Amsterdam.

Lea, C.-D. and M.L. Puri (1988). Asymptotic properties of linear functions of order statistics. J. Statist.

Plann. Inference 18, 203-223.

Ledoux, M. and M. Talagrand (1986). La loi du logarithme it&r6 dans les espaces de Banach. C.R. Acad.

Sci. Paris 303, 57-60.

Pisier, G. (1975). Le theoreme de la limite et la loi du logarithme it&e dans les espace de Banach.

Se’minaire Maurey-Schwartz 1975-76, Exposes 3 et 4. icole Polytechnique, Paris.

Page 13: Asymptotic behaviour of linear combinations of functions of order statistics

R. Norvaifa, R. Zitikis / Linear combinations of functions of order statistics 317

Pisier, G. and J. Zinn (1978). On the limit theorems for random variables with values in the spaces L,

(25p<m). Z. Wuhrsch. Verw. Gebiete 41, 289-304.

Serfling, R.J. (1980). Approximation Theorems of Mathematical Statistics. John Wiley, New York.

Shorack, G.R. (1972). Functions of order statistics. Ann. Math. Sfatist. 43, 412-427.

Singh, K. (1981). On asymptotic representation and approximation to normality of L-statistics - I.

Sankhyd Ser. A 43, 67-83.

Stigler, S.M. (1974). Linear functions of order statistics with smooth weight functions. Ann. Star&/. 2,

676-693.

Vandemaele, M. and N. Veraverbeke (1982). Cramer type large deviations for linear combinations of

order statistics. Ann. Probab. 10, 423-434.