A Best Possible Improvement of Wald's Equation

Abstract

Let $X_1, X_2,\ldots$ be independent random elements taking values in a Banach space $(B, \|\cdot\|)$ and having partial sums $S_n = X_1 + \cdots + X_n$. Let $\alpha > 0$ and let $\Phi: \lbrack 0, \infty)$ be a nondecreasing continuous function such that $\Phi(0) = 0$ and $\Phi(cx) \leq c^\alpha\Phi(x)$ for all $c \geq 2, x \geq 0$. Put $a^\ast_n = E \max_{1\leq k\leq n}\Phi(\|S_k\|)$. Let $T$ be any (possibly randomized) stopping time w.r.t. $\{S_n\}$. We prove that $E \max_{1\leq n\leq T}\Phi(\|S_n\|) \leq 20(18^\alpha)Ea^\ast_T$. If $\{S_n\}$ is a mean-zero $B$-valued martingale and $\lim_{n\rightarrow\infty}E\|S_{T\wedge n}\| < \infty$, it is shown that $L \equiv \lim_{n\rightarrow\infty}ES_nI(T > n)$ always exists and $ES_T = -L$, so that $ES_T = 0 \operatorname{iff} L = 0$. Let $s_n = E\|S_n\|$ and $s^\ast_n = E \max_{1\leq k\leq n}\|S_k\|$. As a consequence of these facts it follows that if $\{X_n\}$ are independent and have mean zero, then $E\|S_T\| < \infty$ and $ES_T = 0$ whenever $Es^\ast_T < \infty$. In the mean-zero case $s^\ast_n \leq 4s_n$; and so, in fact, $Es_T < \infty$ implies $ES_T = 0$. This constitutes a best possible improvement of Wald's equation.