On the Supremum of $S_n/n$

Abstract

Let $X_1, X_2, \cdots$ be independent and identically distributed. We give a simple proof based on stopping times of the known result that $\sup(|X_1 + \cdots + X_n|/n)$ has a finite expected value if and only if $E|X| \log |X|$ is finite. Whenever $E|X| \log |X| = \infty$, a simple nonanticipating stopping rule $\tau$, not depending on $X$, yields $E(|X_1 + \cdots + X_\tau|/\tau) = \infty$.