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


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