On the Law of the Iterated Logarithm

Abstract

Let $X_1, X_2,\cdots$ be a sequence of independent random variables, each with zero mean and finite variance. Define $S_n = \sum^n_{i=1} X_i, s_n^2 = E(S_n^2), t_n^2 = 2 \log \log s_n^2$ and $\Lambda = \lim \sup_{n\rightarrow\infty} S_n/(s_n t_n)$. Suppose that $|X_n| \leqq c_n s_n$ a.s. for all $n$ and some real sequence $\{c_n\}$ and that $s_n \rightarrow \infty$, and let $\nu = \lim \sup_{n\rightarrow\infty} t_nc_n$. By Kolmogorov's law of the iterated logarithm, $\Lambda = 1$ if $\nu = 0$. Egorov proved that $0 \leqq \Lambda < \infty$, in every case. In this paper it will be shown that, if $\nu < \infty$, then $0 < \Lambda \leqq 1 + \sum^\infty_{j=3} \nu^{j-2}/j!$. A similar result for certain classes of unbounded random variables will also be presented.