On the Law of the Iterated Logarithm for L-statistics without Variance

Let {X,Xn; n ≥ 1} be a sequence of i.i.d. random variables with distribution function F (x). For each positive integer n, let X1:n ≤ X2:n ≤ · · · ≤ Xn:n be the order statistics of X1, X2, · · · , Xn. Let H(·) be a real Borel-measurable function defined 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) = E(J(U)H(F←(U))) and Ln(F, J,H) = 1 n ∑n i=1 J ( i n ) H(Xi:n), n ≥ 1, where U is a random variable with the uniform (0, 1) distribution and F←(t) = inf{x; F (x) ≥ t}, 0 < t < 1. In this note, the Chung-Smirnov LIL for empirical processes and the EinmahlLi LIL for partial sums of i.i.d. random variables without variance are used to establish necessary and sufficient conditions for having with probability 1: 0 < lim supn→∞ √ n/φ(n) |Ln(F, J,H)− μ| < ∞, where φ(·) is from a suitable subclass of the positive, nondecreasing, and slowly varying functions defined on [0, ∞). The almost sure value of the limsup is identified under suitable conditions. Specializing our result to φ(x) = 2(log log x), p > 1 and to φ(x) = 2(log x), r > 0, we obtain an analog of the HartmanWintner-Strassen LIL for L-statistics in the infinite variance case. A stability result for L-statistics in the infinite variance case is also obtained. Received October 11, 2007 and in revised form January 8, 2008. AMS Subject Classification: Primary 60F15; Secondary 62G30.