The Finite Mean LIL Bounds are Sharp

Abstract

Let $X, X_1, X_2, \cdots$ be i.i.d. nonconstant mean zero random variables and put $S_n = X_1 + \cdots + X_n$. Let $K(y) > 0$ satisfy $yE\{|X/K(y)|^2 \wedge |X/K(y)|\} = 1$ (for $y > 0$). Then let $a_n = (\log \log n)K(n/\log \log n)$ and $L = \lim \sup_{n\rightarrow\infty}S_n/a_n.$ It is known that $L$ is finite iff $P(X_n > a_n \text{i.o.}) = 0$. When $L < \infty$, it is also known that $1 \leq L \leq 1.5$ and that it is possible for $L$ to equal one. In this paper we construct an example for which $L = 1.5$.