An Extended Version of the Erdos-Renyi Strong Law of Large Numbers

Abstract

Consider a sequence $X_1, X_2, \ldots$, of i.i.d. random variables. For each integer $m \geq 1$ let $S_m$ denote the $m$th partial sum of these random variables and set $S_0 = 0$. Assuming that $EX_1 \geq 0$ and the moment generating function $\phi$ of $X_1$ exists in a right neighborhood of 0 the Erdos-Renyi strong law of large numbers states that whenever $k(n)$ is a sequence of positive integers such that $\log n/k(n) \sim c$ as $n \rightarrow \infty$, where $0 < c < \infty$ then $\max\{(S_{m + k(n)} - S_m)/(\gamma(c)k(n)): 0 \leq m \leq n - k(n)\}$ converges almost surely to 1, where $\gamma(c)$ is a constant depending on $c$ and $\phi$. An extended version of this strong law is presented which shows that it remains true in a slightly altered form when $\log n/k(n) \rightarrow \infty$.