## A best possible improvement of Wald’s equation.(English)Zbl 0648.60050

Let $$X_ 1,X_ 2,..$$. be independent random variables taking values in a Banach space $$(B,\| \cdot \|)$$ and let $$S_ n=X_ 1+...+X_ n$$, $$\alpha_ n=E \| S_ n\|$$. The main theorem of the paper states that if the $$X_ n's$$ have zero mean then for any (possibly randomized) stopping time T w.r.t. $$\{X_ n\}_ n$$, the condition: $$E \alpha_ T<\infty$$ implies $$E S_ T=0$$. Moreover, if $$\lim_{n}E \| S_{T\wedge n}\| <\infty$$ then $$E S_ T=-\lim_{n}E S_ nI_{\{T>n\}}.$$ This result is obtained with the help of the following moment estimate: Let $$X_ n's$$ be as before, but without any assumption on the mean. For any $$\alpha >0$$ there exists a universal constant $$C(\alpha)$$ such that if $$\Phi: R_+\to R_+$$ is a continuous nondecreasing function, $$\Phi (0)=0$$, $$\Phi$$ (cx)$$\leq c^{\alpha} \Phi (x)$$ for all $$c\geq 2$$, $$x\geq 0$$, then $E\max_{1\leq n\leq T}\Phi (\| S_ n\|) \leq C(\alpha)E \alpha^*_ T,\text{ where } \alpha^*_ n=E\max_{1\leq k\leq n}\Phi (\| S_ k\|),$ and T is any stopping time.
Reviewer: T.Bojdecki

### MSC:

 60G40 Stopping times; optimal stopping problems; gambling theory 60G50 Sums of independent random variables; random walks 60E15 Inequalities; stochastic orderings

### Keywords:

Wald’s equation; stopping time
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