On Stopping Rules and the Expected Supremum of $S_n/a_n$ and $|S_n|/a_n$

Abstract

Let $\{X_n\}$ be a sequence of i.i.d. mean zero random variables. Let $S_n = X_1 + \cdots + X_n$. This paper is devoted to determining the conditions where-by $E\sup_{n\geqq 1}S_n/a_n < \infty$ and $E\sup_{n\geqq 1}|S_n|/a_n < \infty$ for quite general sequences of increasing constants $\{a_n\}$. For the sequences $\{a_n\}$ considered, we find it sufficient to examine whether or not $\lim_{n\rightarrow\infty} E(\sum^n_{k=1}X_k/a_k)^+ < \infty$. The existence of optimal extended-valued stopping rules with finite expected reward for sequences $\{S_n/a_n\}$ or $\{|S_n|/a_n\}$ is a by-product of our results. This generalizes results of D. L. Burkholder, Burgess Davis, R. F. Gundy, B. J. McCabe and L. A. Shepp, who treat the case $a_n = n$.