## Wald’s equation for a class of denormalized $$U$$-statistics.(English)Zbl 0784.60039

For the sequence $$\{X_ n\}$$ of zero mean, i.i.d. random variables define $$S_{k,n}=\sum'X_{i_ 1}\ldots X_{i_ k}$$, where $$\sum'$$ denotes the sum over $$1 \leq i_ 1<\cdots <i_ k \leq n$$. Let $$T$$ be a stopping time of $$\{X_ n\}$$. This paper proves that if $$E| X^ p_ 1 |<\infty$$, $$p \in(1,2]$$, and $$ET^{(k-1)/(p-1)}<\infty$$, $$k>1$$, then the Wald-type equation $$ES_{k,T}=0$$ holds. This result complements the result for the case $$k=1$$ given by Y. S. Chow, H. Robbins and D. Siegmund [Great expectations: The theory of optimal stopping (1971; Zbl 0233.60044)]. The result is then utilized to study the moments of $$T_ k=\inf \{n \geq k:\;S_{k,n} \geq 0\}$$ and $$W_ c= \inf \{n \geq 2:\;S^ 2_{1,n} \geq c \sum^ n_{j=1}X^ 2_ j\}$$, $$c>0$$.
Reviewer: N.Weber (Sidney)

### MSC:

 60F99 Limit theorems in probability theory 60G40 Stopping times; optimal stopping problems; gambling theory

### Keywords:

Wald’s equation; $$U$$-statistics; stopping time

Zbl 0233.60044
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