On the Existence of the Optimal Stopping Rule in the $S_n/n$ Problem When the Second Moment is Infinite

Abstract

Let $X_1, X_2, \cdots$ be i.i.d. random variables with mean 0, and let $S_n = \sum^n_{i=1} X_i$. The $S_n/n$ optimal stopping problem is to maximize $E(S_\tau/\tau)$ among finite-valued stopping times $\tau$ relative to the process $(S_n, n \geqq 1)$. In this paper we prove partially Dvoretzky's (1967) conjecture that an optimal stopping time should exist when $E|X_1|^\beta < \infty$ for some $\beta > 1$, by showing that the result holds if $\lim \sup_{n\rightarrow\infty} P(S_n \geqq c\|S_n\|) > 0$ for some $c > 0$, where $\|S_n\| = (E|S_n|^\beta)^{1/\beta}$. This condition is shown to hold in some special cases, including the case where the $X_i$ are in the domain of attraction of a stable distribution with exponent greater than one.