Properties of Optimal Extended-Valued Stopping Rules for $S_n/n^1$

Abstract

Let $X_1, X_2, \cdots$ be i.i.d. nondegenerate mean zero random variables with common distribution function $F$ such that $EX_1^+ \log^+ X_1 < \infty$. Let $S_n = X_1 + \cdots + X_n$ and $T_\infty$ be the collection of randomized extended-valued stopping rules. For $t\in T_\infty$, define $E(S_t/t) = E(S_t/t)1_{\{t < \infty\}}$. There exists $\tau\in T_\infty$ such that $E(S_\tau/\tau) = \sup_{t\in T_\infty} E(S_t/t) < \infty$. We wish to determine whether or not $P(\tau = \infty) > 0$. Let $b^{-1}(x) = 1/(\int^\infty_x (y/x)\log (y/x) dF(y))$. We find that $P(X_n \geqq b(n) \mathrm{i.o.}) = 1 \Rightarrow P(\tau < \infty) = 1$. As a corollary, if $E(X_1^+)^\alpha < \infty$ for some $\alpha > 1, P(\tau < \infty) = 1$. Conversely, if $P(|X_n| \geqq b(n) \mathrm{i.o.}) = 0$, then $P(\tau = \infty) > 0$. Moreover, if $g(n)\nearrow$ and $\sum^\infty_{n=1} g(2^n)^{-1} < \infty$ then $P(|X_n| \geqq b(n)g(n)\mathrm{i.o.}) = 1 \Rightarrow P(\tau < \infty) = 1$. Examples satisfying these latter conditions are given. An outgrowth of this work is that for any i.i.d. sequence $\{X_n\}$ of mean zero random variables and $c < \frac{1}{4}, P(S_n \geqq cE|S_n|\mathrm{i.o.}) = 1$. The importance of this result stems from the fact that we may also have $S_n(\log n)^\varepsilon/n \rightarrow -\infty$ in probability (see [15]). In order to be completely rigorous, a section was included which provides a useful characterization of the general form of the class of optimal rules for sequential decision problems.