Bulletin of the Institute of Mathematics Academia Sinica (New ) Vol. 3 (2008), No. 3, pp. 417-432

ON THE LAW OF THE ITERATED FOR L- WITHOUT

BY

DELI LI, DONG LIU AND ANDREW ROSALSKY

Abstract

Let {X, Xn; n ≥ 1} be a sequence of i.i.d. random vari- ables with distribution F (x). For each positive inte-

ger n, let X1:n ≤ X2:n ≤ ··· ≤ Xn:n be the order statistics of

X1, X2, · · · , Xn. Let H(·) be a real Borel-measurable function de- fined on R such that E|H(X)| < ∞ and let J(·) be a Lipschitz function of order one defined on [0, 1]. Write µ = µ(F,J,H) = ← 1 n i E L : (J(U)H(F (U))) and n(F,J,H) = n Pi=1 J n  H(Xi n), n ≥ 1, where U is a with the uniform (0, 1) dis- ← tribution and F (t) = inf{x; F (x) ≥ t}, 0

Received October 11, 2007 and in revised form January 8, 2008. AMS Subject Classification: Primary 60F15; Secondary 62G30. Key words and phrases: Law of the iterated logarithm; strong ; L-statistics.

417 418 DELILI,DONGLIUANDANDREWROSALSKY [September

1. Introduction

Let X, X ; n 1 be a sequence of independent and identically dis- { n ≥ } tributed (i.i.d.) real random variables with distribution function F (x) = P (X x), x = ( , ) and let U, U ; n 1 be a sequence of i.i.d. ≤ ∈ R −∞ ∞ { n ≥ } random variables with the uniform (0, 1) distribution. For each positive inte- ger n, let X X X be the order statistics of X , X , , X . 1:n ≤ 2:n ≤···≤ n:n 1 2 · · · n Let H( ) be a real-valued Borel-measurable function defined on . A linear · R combination of order statistics (in short, an L-statistic) is a statistic of the form n 1 L = c H(X ) n n i,n i:n Xi=1 where the weights c , 1 i n are real numbers and n 1. Define i,n ≤ ≤ ≥ Lt = log max e, t and LLt = L(Lt) for t . The classical Hartman- e { } ∈ R Wintner-Strassen law of the iterated logarithm (LIL) states that

n H(X ) + lim sup(lim inf) i=1 i =( ) σ (a.s.) (1.1) →∞ n→∞ √ − n P 2nLLn if and only if EH(X)=0 and σ2 = EH2(X) < . (1.2) ∞ Moreover, if (1.2) holds, then

n C H(Xi)/√2nLLn; n 1 = [ σ, σ] a.s., (1.3) ( ≥ )! − Xi=1 where C ( x ; n 1 ) stands for the cluster set (i.e., the set of limit points) { n ≥ } of the numerical sequence x ; n 1 . See Hartman and Wintner (1941) { n ≥ } for the “if” part and Strassen (1966) for the converse. The conclusion (1.3) is due to Strassen (1964). Alternative proofs of the Hartman-Wintner (1941) LIL were discovered by Strassen (1964), Heyde (1969), Egorov (1971), Teicher (1974), Cs¨org˝o and R´ev´esz (1981, p. 119), de Acosta (1983), Li (1987), and Einmahl and Li (2005). Substantially simpler proofs of Strassen’s (1966) converse were ob- tained by Feller (1968), Heyde (1968), and Steiger and Zaremba (1972). Mar- tikainen (1980), Rosalsky (1980), and Pruitt (1981) simultaneously and inde- 2008] LIL FOR L-STATISTICS 419 pendently obtained a “one-sided” converse to the Hartman-Wintner (1941) LIL. Specifically, they proved that each part of (1.1) individually implies (1.2). Many authors, including Helmers (1977), Helmers, Janssen, and Serfling (1988), Li, Rao, and Tomkins (2001), Mason (1982), Sen (1978), van Zwet (1980), and Wellner (1977a,b), have investigated the strong limiting behavior for a class of L-statistics of the form

n 1 i L (F,J,H) J H(X ), n 1 (1.4) n ≡ n n i:n ≥ Xi=1   where J( ) is a real-valued function, often called a score function, defined · on [0, 1]. Helmers (1977), Mason (1982), Sen (1978), van Zwet (1980), and Wellner (1977a) have studied the strong law of large numbers (SLLN) for L , n 1 and have shown that under a variety of conditions on J( ) and n ≥ · H( ) · n 1 i 1 lim Ln(FU ,J,H)= lim J H(Ui:n)= J(t)H(t)dt (finite) a.s., →∞ →∞ n n n n 0 Xi=1   Z where F is the distribution function of the random variable U and U , 1 U i:n ≤ i n, are the order statistics of the U , 1 i n, n 1. ≤ i ≤ ≤ ≥ If J( ) is a Lipschitz function of order one defined on [0, 1] and · E H(X) < , (1.5) | | ∞ let us write

Z = J(U)H(F ←(U)), (1.6) Y = Z + µ 1 (I(U t) t) J ′(t)H(F ←(t))dt, ( − − 0 ≤ − R where F ←(t) is the quantile function

F ←(t) = inf s; F (s) t , 0

Then µ exists and is finite and Y and Z are both well-defined random vari- ables under (1.5). Moreover,

2 E 2 σY = Var(Y )= Y . (1.8)

To see this, note that (1.8) is equivalent to

1 EY = E (I(U t) t) J ′(t)H(F ←(t))dt = 0. (1.9) − ≤ − Z0  Since E (I(U t) t) = 0, (1.9) follows from an application of Fubini’s ≤ − , subject to the existence of the integral

1 1 I = J ′(t)H(F ←(t))dt = H(F ←(t))dJ(t)= E J ′(U)H(F ←(U)) . Z0 Z0  But the score function J( ) has an (with respect to · Lebesgue measure) bounded derivative J ′( ). From this fact and the equality · E H(X) = E H(G(U)) < , it follows that I exists and is finite; clearly, | | | | ∞ σ2 < if and only if Y ∞ EZ2 < . (1.10) ∞

Recall that a sequence of random variables ξ ; n 1 is said to be { n ≥ } bounded in probability if

lim sup P ( ξn x) = 0. x→∞ n≥1 | |≥

Combining the Chung-Smirnov LIL (see Chung (1949) and Smirnov (1944)) and the Finkelstein functional LIL (see Finkelstein (1971)) for empirical pro- cesses, Li, Rao, and Tomkins (2001, Theorem 2.1) proved that the following three statements are equivalent:

(1.10) holds;

lim sup n/(2LLn) Ln(F,J,H) µ < a.s.; n→∞ | − | ∞ p √n (L (F,J,H) µ) ; n 1 is bounded in probability. n − ≥  2008] LIL FOR L-STATISTICS 421

Moreover, if any of the three statements above holds, then

+ lim sup(lim inf) n/(2LLn) (Ln(F,J,H) µ)=( ) σY a.s., (1.11) n→∞ n→∞ − − p

C n/(2LLn) (L (F,J,H) µ) ; n 1 = [ σ , σ ] a.s., (1.12) n − ≥ − Y Y np o and d √n (L (F,J,H) µ) N(0, σ2 ), (1.13) n − −→ Y where “ d ” denotes convergence in distribution. This powerful result con- −→ tains many previous results obtained under more restrictive conditions, al- though it is still not the last word as the authors mention in an open problem (to weaken the conditions on J( )). The authors illustrate with examples · that their result can handle some cases that previous results could not; for example, the Gini -difference statistic.

The main purpose of the present note is to find necessary and sufficient conditions for

0 < lim sup n/ϕ(n) Ln(F,J,H) µ < a.s., n→∞ | − | ∞ p where ϕ( ) is from a suitable subclass of the positive, nondecreasing, and · slowly varying functions. But we also treat the case where the limit is 0 a.s. We emphasize that we are not assuming that EZ2 < where Z is as in ∞ (1.6).

The plan of this note is as follows. Our main results, 2.1 and 2.2, and their proofs and corollaries are presented in Section 2. The proofs are obtained via a nice application of the Chung-Smirnov LIL (see Chung (1949) and Smirnov (1944)) for empirical processes and the Einmahl-Li LIL (see Einmahl and Li (2005)) for partial sums of i.i.d. random variables with- out variance. This work of Einmahl and Li (2005) is related to previous work of Martikainen (1984); it appears that slightly different proofs of Theorems 2.1 and 2.2 can be given using this work of Martikainen (1984) instead of 422 DELILI,DONGLIUANDANDREWROSALSKY [September that of Einmahl and Li (2005). In Section 3, we provide an example to illustrate our results.

2. Main Results

Let be the set of continuous, nondecreasing functions ϕ( ) : [0, ) H · ∞ → (0, ), which are slowly varying at infinity. By monotonicity, the slow ∞ variation of ϕ( ) is equivalent to lim →∞ ϕ(et)/ϕ(t) = 1. Very often one · t can even show that limt→∞ ϕ(tf(t))/ϕ(t) = 1, where f( ) is a nondecreasing · τ function such that lim →∞ f(t) = . Set f (t) = exp((Lt) ), 0 τ < 1. t ∞ τ ≤ Given 0 q < 1, let be the class of functions ϕ( ) such that ≤ Hq ⊂H · ϕ (tf (t)) lim τ = 1, 0 τ < 1 q t→∞ ϕ(t) ≤ − and set = . H1 H We consider q to be a measure for how slow the slow variation is. So functions in are the “slowest” and it will turn out that this class is par- H0 ticulary interesting for LIL type results (see Theorem 2.2 below). Examples of functions in are ϕ(t) = (Lt)r, r 0 and ϕ(t) = (LLt)p, p 0. H0 ≥ ≥ The following Theorem 2.1 gives LIL type results when λ > 0 and stability results when λ = 0 with respect to a large class of normalizing sequences, without assuming that EZ2 < , where Z is defined in (1.6). ∞ Theorem 2.1. Let X, X ; n 1 be a sequence of i.i.d. random vari- { n ≥ } ables with distribution function F (x)= P (X x), x and let H( ) be a ≤ ∈ R · real-valued Borel-measurable function defined on satisfying (1.5). Let J( ) R · be a Lipschitz function of order one defined on [0, 1] and let L (F,J,H), n n ≥ 1, Z, and µ be defined by (1.4), (1.6), and (1.7), respectively. Given a func- tion ϕ( ) where 0 q 1, set Ψ(x)= xϕ(x),x . If · ∈Hq ≤ ≤ ∈ R log log x p lim = (2.1) x→∞ ϕ(x) ∞ and

Ψ−1(xLLx) E −1 E 2 Ψ ( Z ) < and λ = 2limsup 2 (Z I Z x ), (2.2) | | ∞ s x→∞ x LLx {| |≤ } 2008] LIL FOR L-STATISTICS 423 then when λ< we have ∞ 1/2 (1 q) λ lim sup n/ϕ(n) Ln(F,J,H) µ λ a.s. (2.3) − ≤ n→∞ | − |≤ p Conversely, if q < 1, then the relation

lim sup n/ϕ(n) Ln(F,J,H) µ < a.s. (2.4) n→∞ | − | ∞ p implies that (2.2) holds with λ< . ∞ Moreover, the limsup in (2.3) is positive and finite if and only if (2.2) holds with 0 <λ< . ∞ For slowly varying functions ϕ( ) and λ as in (2.2) with 0 <λ< · ∈ H0 , we obtain for L-statistics L (F,J,H), n 1 the following complete ∞ { n ≥ } analogue of the Hartman-Wintner-Strassen LIL. Of course, (2.5) follows from (2.6) but nevertheless it is worthwhile to label them separately.

Theorem 2.2. Assume that H( ) is a real-valued Borel-measurable · function defined on satisfying (1.5) and that ϕ( ) satisfies (2.1). R · ∈ H0 Assume that X, X ; n 1 , J( ), L (F,J,H), Z, µ, and Ψ( ) are as in { n ≥ } · n · Theorem 2.1. Let 0 λ< . Then ≤ ∞ + lim sup(lim inf) n/ϕ(n) (Ln(F,J,H) µ)=( ) λ a.s. (2.5) n→∞ n→∞ − − p and

C n/ϕ(n) (L (F,J,H) µ) ; n 1 = [ λ, λ] a.s. (2.6) n − ≥ − np o if and only if condition (2.2) holds.

Remark 2.1. Due to condition (2.1), Theorems 2.1 and 2.2 do not include as a special case the classical Hartman-Wintner-Strassen LIL for L-statistics obtained by Li, Rao, and Tomkins (2001, Theorem 2.1). It is interesting to note that, under the condition (2.1), the limiting behavior in Theorems 2.1 and 2.2 is determined by the distribution of Z, whereas, by contrast, the three conclusions (1.11), (1.12), and (1.13) depend on the distribution of Y , where Y is defined in (1.6). 424 DELILI,DONGLIUANDANDREWROSALSKY [September

Remark 2.2. In general, it is not easy to find F ←(t), 0

Remark 2.3. We conjecture that Theorems 2.1 and 2.2 are still true without condition (1.5).

Remark 2.4. We note that if EZ2 < , then the constant λ in Theo- ∞ rems 2.1 and 2.2 is simply

Ψ−1(xLLx) E 2 λ = 2( Z )limsup 2 . s x→∞ x LLx

We shall illustrate Theorem 2.2 by considering the following two special cases:

Case I. Choose ϕ(x) = 2(LLx)p where p> 1. Then one can check that

Ψ−1(xLLx)/(x2LLx) lim = 1. x→∞ 1/(2(LLx)p−1)

Case II. Take ϕ(x) = 2(Lx)r where r > 0. Then one also easily sees that Ψ−1(xLLx)/(x2LLx) lim = 1. x→∞ LLx/(Lx)r Thus, Theorem 2.2 yields the following two results.

Corollary 2.1. Assume that H( ) is a real-valued Borel-measurable · function defined on satisfying (1.5). Let p> 1. For any constant 0 λ< R ≤ , we have: ∞ + n L lim sup(lim inf) p ( n(F,J,H) µ)=( ) λ a.s. n→∞ n→∞ 2(LLn) − − r 2008] LIL FOR L-STATISTICS 425 and n C (L (F,J,H) µ) ; n 1 = [ λ, λ] a.s. 2(LLn)p n − ≥ − r  if and only if

Z2 E 1−pE 2 p < and λ = lim sup(LLx) (Z I Z x ). (LL Z ) ∞ x→∞ {| |≤ }  | |  r

Corollary 2.2. Assume that H( ) is a real-valued Borel-measurable · function defined on satisfying (1.5). Let r> 0. For any constant 0 λ< R ≤ , we have: ∞ + n L lim sup(lim inf) r ( n(F,J,H) µ)=( ) λ a.s. n→∞ n→∞ 2(Ln) − − r and n C (L (F,J,H) µ) ; n 1 = [ λ, λ] a.s. 2(Ln)r n − ≥ − r  if and only if

Z2 LLx E −r E 2 r < and λ = 2 lim sup r (Z I Z x ). (L Z ) ∞ x→∞ (Lx) {| |≤ }  | |  s

If condition (2.2) is satisfied with λ = 0, we obtain the following stability result for L-statistics.

Corollary 2.3. Assume that H( ) is a real-valued Borel-measurable · function defined on satisfying (1.5). Let ϕ( ) and let Ψ( ) be as in R · ∈ H · Theorem 2.1. If

Ψ−1(xLLx) EΨ−1( Z ) < and lim E Z2I Z x = 0, (2.7) | | ∞ x→∞ x2LLx {| |≤ }  then

lim n/ϕ(n) (Ln(F,J,H) µ) = 0 a.s. (2.8) n→∞ − p 426 DELILI,DONGLIUANDANDREWROSALSKY [September

Moreover, if ϕ( ) for some 0 q < 1, then condition (2.7) is necessary · ∈Hq ≤ and sufficient for (2.8) to hold.

Proof of Theorems 2.1 and 2.2. Let U, U ; n 1 represent a { n ≥ } sequence of i.i.d. random variables with the uniform (0, 1) distribution. Then it is well known that

d X, X ; n 1 = F ←(U), F ←(U ); n 1 . { n ≥ } { n ≥ } It now follows that

d X ; 1 i n,n 1 = F ←(U ); 1 i n,n 1 , { i:n ≤ ≤ ≥ } { i:n ≤ ≤ ≥ } where Ui:n, 1 i n, are the order statistics of Ui, 1 i n. Thus, one ≤ ≤ ← ≤ ≤ ← may set without loss of generality Xn = F (Un) and Xi:n = F (Ui:n) for 1 i n and n 1. Note that ≤ ≤ ≥ P (U = U for all 1 i

n i J( )H (X ) n i:n i=1 X n i = J( )H (F ←(U )) n i:n i=1 Xn ← = J(Ui:n)H (F (Ui:n)) i=1 Xn i + J( ) J(U ) H (F ←(U )) (2.9) n − i:n i:n i=1   Xn a.s. ← = J(Ui)H (F (Ui)) i=1 Xn + (J(D (U )) J(U )) H (F ←(U )) n i:n − i:n i:n i=1 n X = J(U )H (F ←(U )) + R (say), n 1 i i n ≥ Xi=1 where D (t) n−1 n I U t is the empirical distribution function n ≡ i=1 { i ≤ } of U ,U , ..., U . Since J( ) is a Lipschitz function of order one defined on 1 2 n P · 2008] LIL FOR L-STATISTICS 427

[0, 1], there exists a constant 0 C < , depending on J( ) only, such that ≤ ∞ · J(t ) J(t ) C t t uniformly for t , t [0, 1]. Hence | 1 − 2 |≤ | 1 − 2| 1 2 ∈ n ← Rn C max Dn(Ui:n) Ui:n H(F (Ui:n)) | | ≤ 1≤i≤n | − | | | i=1 n X ← C sup Dn(t) t H(F (Ui)) , n 1. ≤ 0≤t≤1 | − | | | ≥ Xi=1 Since (1.5) holds, i.e., E H(F ←(U)) = E H(X) < , by the Kolmogorov | | | | ∞ SLLN, we have

n H(F ←(U )) lim i=1 | i | = E H(X) a.s. n→∞ n | | P Note that the Chung-Smirnov LIL for empirical processes (see Chung (1949) and Smirnov (1944)) states that

n 1 lim sup sup Dn(t) t = a.s. →∞ 2LLn | − | 2 n r 0≤t≤1 Thus, for any given ϕ( ) satisfying (2.1), since · ∈H R n 2LLn n H(F ←(U )) | n| C sup D (t) t i=1 | i |, n 1, ≤ 2LLn | n − |× ϕ(n) n ≥ nϕ(n) r 0≤t≤1 s P wep have R lim n = 0 a.s. (2.10) n→∞ nϕ(n) It then follows from (2.9) and (2.10)p that, for any given ϕ( ) satisfying · ∈H (2.1),

n −1 ← lim n/ϕ(n) Ln(F,J,H) n J(Ui)H (F (Ui)) = 0 a.s. (2.11) n→∞ − i=1 ! p X Ψ−1(xLLx) It is easy to see that ϕ( ) and (2.1) imply that 2 is slowly · ∈ Hq x LLx varying at infinity with

Ψ−1(xLLx) lim = 0. x→∞ x2LLx 428 DELILI,DONGLIUANDANDREWROSALSKY [September

Thus (2.2) with λ< is equivalent to ∞ −1 2 E −1 Ψ (xLLx)E 2 λ Ψ ( V ) < and lim sup 2 V I Z x = (2.12) | | ∞ x→∞ x LLx {| |≤ } 2  where V = J(U)H(F ←(U)) µ = Z µ. Then, by Theorem 1 of Einmahl − − and Li (2005), (2.12) implies that

n J(U )H(F ←(U )) (1 q)1/2λ lim sup n/ϕ(n) i=1 i i µ λ a.s. − ≤ →∞ n − ≤ n P p (2.13)

and (2.3) follows from (2.11) and (2.13).

Conversely, by Theorem 1 of Einmahl and Li (2005), if 0 q < 1, then ≤ the relation n J(U )H(F ←(U )) lim sup n/ϕ(n) i=1 i i µ < a.s. (2.14) →∞ n − ∞ n P p implies that (2.12) holds with λ< and, moreover, the lim sup in (2.14) is ∞ positive if and only if (2.12) holds with 0 <λ< . Thus combining (2.4) ∞ and (2.11) yields (2.14) and hence (2.12) holds with λ < . As was noted ∞ above, (2.2) with λ < is equivalent to (2.12), and the last assertion in ∞ Theorem 2.1 is now immediate.

Similarly, if ϕ( ) and (2.1) holds then by combining Theorem 2 of · ∈H0 Einmahl and Li (2005) and (2.11), the proof of Theorem 2.2 follows. 

3. An Interesting Example

In this section, we shall provide an example to illustrate our results.

Example. Take J(t) = 4t 2, 0 t 1, and H(x)= x, x . Then − ≤ ≤ ∈ R the L-statistic n 1 i L (F,J,H)= 4 2 X n n · n − i:n Xi=1   2008] LIL FOR L-STATISTICS 429 is related to Gini’s mean difference,

n 2 1 i 1 X X = 4 − 2 X , n(n 1) i j n n 1 i:n ≤ ≤ | − | · − − 1 Xi

1/2 n 2 (1 q) λ lim sup Xi Xj θ λ a.s. − ≤ n→∞ ϕ(n) n(n 1) | − |− ≤ 1≤i

Conversely, if q < 1, then the relation

n 2 lim sup Xi Xj θ < a.s. (3.2) n→∞ ϕ(n) n(n 1) | − |− ∞ 1≤i

implies that (3.1) holds with λ < . Moreover, the lim sup in (3.2) is ∞ 430 DELILI,DONGLIUANDANDREWROSALSKY [September positive if and only if (3.1) holds with 0 <λ< . If q = 0, then for any ∞ constant 0 λ< , ≤ ∞

n 2 + lim sup(lim inf) Xi Xj θ =( ) λ a.s. →∞ n→∞ ϕ(n) n(n 1)  n ≤ ≤ | − |− − r − 1 Xi

n 2 C X X θ ; n 1 = [ λ, λ] a.s.  ϕ(n) n(n 1) i j   ≤ ≤ | − |− ≥ − r − 1 Xi

Acknowledgments

The research of the first author was supported by a grant from the Natural Sciences and Engineering Research Council of Canada.

References

1. A. de Acosta, A new proof of the Hartman-Wintner law of the iterated logarithm, Ann. Probab., 11(1983), 270-276. 2. K. L. Chung, An estimate concerning the Kolmogoroff limit distribution, Trans. Amer. Math. Soc., 67(1949), 36-50. 3. M. Cs¨org˝oand P. R´ev´esz, Strong Approximations in Probability and Statistics, Academic Press, New York, 1981. 4. V. A. Egorov, A generalization of the Hartman-Wintner theorem on the law of the iterated logarithm, Vestnik Leningrad. Univ., 1971 (1971), 22-28 (in Russian). English translation in Vestnik Leningrad Univ. Math., 4 (1977), 117-124. 5. U. Einmahl and D. Li, Some results on two-sided LIL behavior, Ann. Probab., 33(2005), 1601-1624. 6. W. Feller, An extension of the law of the iterated logarithm to variables without variance, J. Math. Mech., 18(1968), 343-356. 7. H. Finkelstein, The law of the iterated logarithm for empirical distributions, Ann. Math. Statist., 42(1971), 607-615. 8. P. Hartman and A. Wintner, On the law of the iterated logarithm, Amer. J. Math., 63(1941), 169-176. 2008] LIL FOR L-STATISTICS 431

9. R. Helmers, A strong law of large numbers for linear combinations of order statis- tics, Report SW50/77 (1977), Mathematisch Centrum, Amsterdam, The Netherlands. 10. R. Helmers, P. Janssen, and R. Serfling, Glivenko-Cantelli properties of some gen- eralized empirical DF’s and strong convergence of generalized L-statistics, Probab. Theory Related Fields, 79 (1988), 75-93. 11. C. C. Heyde, On the converse to the iterated logarithm law, J. Appl. Probab., 5(1968), 210-215. 12. C. C. Heyde, Some properties of metrics in a study on convergence to normality, Z. Wahrsch. Verw. Gebiete, 11(1969), 181-192. 13. D. L. Li, A simple proof of the Hartman-Wintner law of the iterated logarithm, Acta Sci. Natur. Univ. Jilin, 1987(1987), 43-45 (in Chinese). 14. D. Li, M. B. Rao, and R. J. Tomkins, The law of the iterated logarithm and central limit theorem for L-statistics, J. Multivariate Anal., 78(2001), 191-217. 15. A. I. Martikainen, A converse to the law of the iterated logarithm for a , Teor. Veroyatnost. i Primenen., 25(1980), 364-366 (in Russian). English translation in Theory Probab. Appl., 25 (1981), 361-362. 16. A. I. Martikainen, Criteria for strong convergence of normalized sums of indepen- dent random variables and their applications, Teor. Veroyatnost. i Primenen., 29 (1984), 502-516 (in Russian). English translation in Theory Probab. Appl., 29(1985), 519-533. 17. D. M. Mason, Some characterizations of strong laws for linear functions of order statistics, Ann. Probab., 10(1982), 1051-1057. 18. W. E. Pruitt, General one-sided laws of the iterated logarithm, Ann. Probab., 9 (1981), 1-48. 19. A. Rosalsky, On the converse to the iterated logarithm law, Sankhy¯aSer. A, 42(1980), 103-108. 20. P. K. Sen, An invariance principle for linear combinations of order statistics, Z. Wahrsch. verw. Gebiete, 42 (1978), 327-340. 21. R. J. Serfling, Approximation Theorems of Mathematical Statistics, John Wiley, New York, 1980. 22. G. R. Shorack and J. A. Wellner, Empirical Processes with Applications to Statis- tics, Wiley, New York, 1986. 23. N. V. Smirnov, An approximation to distribution laws of random quantities de- termined by empirical data, Uspehi Matem. Nauk., 10(1944), 179-206 (in Russian). 24. W. L. Steiger and S. K. Zaremba, The converse of the Hartman-Wintner theorem, Z. Wahrsch. Verw. Gebiete, 22 (1972), 193-194. 25. V. Strassen, An invariance principle for the law of the iterated logarithm, Z. Wahrsch. Verw. Gebiete, 3(1964), 211-226. 26. V. Strassen, A converse to the law of the iterated logarithm, Z. Wahrsch. Verw. Gebiete, 4(1966), 265-268. 432 DELILI,DONGLIUANDANDREWROSALSKY [September

27. H. Teicher, On the law of the iterated logarithm, Ann. Probab., 2 (1974), 714-728. 28. W. R. van Zwet, A strong law for linear functions of order statistics, Ann. Probab., 8 (1980), 986-990. 29. J. A. Wellner, A Glivenko-Cantelli theorem and strong laws of large numbers for functions of order statistics, Ann. Statist., 5 (1977a), 473-480. 30. J. A. Wellner, A law of the iterated logarithm for functions of order statistics, Ann. Statist., 5 (1977b), 481-494.

Department of Mathematical Sciences, Lakehead University, Thunder Bay, Ontario, P7B 5E1, Canada. E-mail: [email protected] Department of Mathematical Sciences, Lakehead University, Thunder Bay, Ontario, P7B 5E1, Canada. Department of Statistics, University of Florida, Gainesville, FL 32611, U.S.A. E-mail: [email protected]fl.edu