Moments of randomly stopped \(U\)-statistics
type: Item
instance of: scholarly article
title: Moments of randomly stopped \(U\)-statistics
zbMATH Open document ID: 0902.60037
author: Tze Leung Lai
author: Victor H. de la Peña
published in: The Annals of Probability
publication date: 1998-12-02T00:00:00Z
review text: Let \(U_n\) be a \(U\)-statistics of the general form \(U_n=\sum_{1\leq i_1<\dots< i_k\leq n} f(X_{i_1},\dots,X_{i_k})\), where \(\{X_n\}\) is a sequence of i.i.d.\ random variables. Let \(T\) be a stopping time with respect to the filtration \(\{F_n\}\), where \(F_n=\sigma(X_1,\dots, X_n)\). The authors obtain sharp bounds for \(E(\max_{k\leq n\leq T}| U_n| ^p)\), \(p\geq 1\), under the condition \(E(f(X_1,\dots,X_k)\mid X_{i_1},\dots,X_{i_h}))\) for all \(\{i_1,\dots,i_h\}\subset\{1,\dots,k\}\), \(h<k\), and, in the case \(1\leq p\leq 2\), under the additional condition of symmetry of \(f\) in its arguments. This is applied to prove the Wald identity \(EU_T=0\), when, for some \(p\in(1,2]\), \(E| f(X_1,\dots,X_k)| ^p<\infty\) and \(ET^\rho<\infty\) with \(\rho=(p^{1/(k-1)}-1)^{-1}\). Several applications are provided.
Mathematics Subject Classification ID: 60G40
Mathematics Subject Classification ID: 60F25
Mathematics Subject Classification ID: 62L12
zbMATH DE Number: 1130495
zbMATH Keywords: Wald identity
zbMATH Keywords: martingales
zbMATH Keywords: stopping times
reviewed by: Vigirdas Mackevičius
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1023481120
OpenAlex ID: W2047830941
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DOI: 10.1214/AOP/1023481120
description: scientific article; zbMATH DE number 1130495