On a Problem of Csorgo and Revesz

Abstract

Suppose $\{X_n\}$ is an i.i.d. sequence of random variables with mean 0, variance 1 and $S_n = \sum^n_{i = 1}X_i$. Let $0 < r < 1$. It is well known that $S_n - W(n) = O((\log n)^{1/r}) \mathrm{a.s}.$ when $Ee^{t_0|X_1|^r} < \infty$ for some $t_0 > 0$, where $\{W(t), t \geq 0\}$ is the standard Wiener process. We prove that $O((\log n)^{1/r})$ cannot be replaced by $o((\log n)^{1/r})$.