Wald's equation for a class of denormalized \(U\)-statistics
type: Item
instance of: scholarly article
title: Wald's equation for a class of denormalized \(U\)-statistics
zbMATH Open document ID: 0784.60039
author: Victor H. de la Peña
author: Yuan-Shih Chow
author: Henry Teicher
published in: The Annals of Probability
publication date: 1993-10-11T00:00:00Z
review text: 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 \(W_ c= \inf \{n \geq 2:\;S^ 2_{1,n} \geq c \sum^ n_{j=1}X^ 2_ j\}\), \(c>0\).
Mathematics Subject Classification ID: 60F99
Mathematics Subject Classification ID: 60G40
zbMATH DE Number: 428675
zbMATH Keywords: Wald's equation
zbMATH Keywords: \(U\)-statistics
zbMATH Keywords: stopping time
reviewed by: Neville Weber
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1176989285
OpenAlex ID: W2036510055
Wikidata QID: Q101201648
DOI: 10.1214/AOP/1176989285
description: scientific article; zbMATH DE number 428675