Wald's equation for a class of denormalized \(U\)-statistics
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 \textit{Y. S. Chow}, \textit{H. Robbins} and \textit{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 \(W_ c= \inf \{n \geq 2:\;S^ 2_{1,n} \geq c \sum^ n_{j=1}X^ 2_ j\}\), \(c>0\).
scientific article; zbMATH DE number 428675