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On the law of the iterated logarithm for L-statistics without variance
by Deli Li   Dong Liu   Andrew Rosalsky

Vol. 3 No. 3 (2008) P.417~P.432

ABSTRACT

Let {$X,X_n; n \ge 1$} be a sequence of i.i.d. random variables with distribution function $F(x)$. For each positive integer $n$, let $X_{1:n} \le X_{2:n} \le \dots \le X_{n:n}$ be the order statistics of $X_1, X_2, \dots, X_n$. Let $H(\cdot)$ be a real Borel-measurable function defined on $\mathcal{R}$ such that $\mathbb{E}|H(X)| < \infty$ and let $J(\cdot)$ be a Lipschitz function of order one defined on $[0, 1]$. Write $\mu = \mu(F, J, H) = \mathbb{E}(J(U)H(F_\gets (U)))$ and $\mathbb{L}_n(F, J, H) = \frac 1n \sum^n_{i=1} J(\frac in)H(X_{i:n}), n \ge 1,$ where $U$ is a random variable with the uniform (0,1) distribution and $F^\gets(t) = \inf \{ x; F(x) \ge t \}$, $0 \lt t \lt 1$. 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 $\sup_{n \to \infty} \sqrt{n/ \varphi(n)} | \mathbb{L}_n(F, J, H) - \mu | < \infty$, where $\varphi(\cdot)$ is from a suitable subclass of the positive, non-decreasing, and slowly varying functions defined on $[0, \infty)$. The almost sure value of the limsup is identified under suitable conditions. Specializing our result to $\varphi(x) = 2(\log \log x)^p , p>1$ and to $\varphi(x) = 2(\log x)\gamma, \gamma > 0$, we obtain an analog of the Hartman-Wintner-Strassen LIL for L-statistics in the infinite variance case. A stability result for L-statistics in the infinite variance case is also obtained.

KEYWORDS
Law of the iterated logarithm, strong law of large numbers, L-statistics

MATHEMATICAL SUBJECT CLASSIFICATION 2010
Primary: 60F15, 62G30

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