ON THE LAW OF THE ITERATED LOGARITHM FOR L-STATISTICS WITHOUT VARIANCE

Let {X, X; n >= 1} be a sequence of i.i.d. random variables with distribution function F(x). For each positive integer n, let X-1:n <= X-2:n <= ... <= X-n:n be the order statistics of X-1, X-2, ... , X-n. Let H(.) be a real Borel-measurable function defined on R such that E vertical bar H(X)vertical bar < infinity and let J(.) be a Lipschitz function of order one defined on [0, 1]. Write mu = mu(F, J, H) = E(J(U)II(F-<-(U))) and L-n (F, J, H) - 1/n Sigma(n)(i=1) J(i/n) II (X-i: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 Einmahl-Li 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 sup(n ->infinity) root n/phi(n)vertical bar L-n (F, J, H) - mu vertical bar < infinity, where phi(.) is from a suitable subclass of the positive, non decreasing, and slowly varying functions defined on [0, infinity). The almost sure value of the limsup is identified under suitable conditions. Specializing our result to phi(x) = 2(log log x)(p), p > 1 and to phi(x) = 2(log x)(r), 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.