Open Access
October 1997 Moments of randomly stopped $U$-statistics
Tze Leung Lai, Victor H. de la Peña
Ann. Probab. 25(4): 2055-2081 (October 1997). DOI: 10.1214/aop/1023481120


In this paper we provide sharp bounds on the $L_p$-norms of randomly stopped $U$-statistics. These bounds consist mainly of decoupling inequalities designed to reduce the level of dependence between the $U$-statistics and the stopping time involved. We apply our results to obtain Wald’s equation for $U$-statistics, moment convergence theorems and asymptotic expansions for the moments of randomly stopped $U$-statistics. The proofs are based on decoupling inequalities, symmetrization techniques, the use of subsequences and induction arguments.


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Tze Leung Lai. Victor H. de la Peña. "Moments of randomly stopped $U$-statistics." Ann. Probab. 25 (4) 2055 - 2081, October 1997.


Published: October 1997
First available in Project Euclid: 7 June 2002

zbMATH: 0902.60037
MathSciNet: MR1487445
Digital Object Identifier: 10.1214/aop/1023481120

Primary: 60F25 , 60G40
Secondary: 62L12

Keywords: $U$-statistics , Decoupling inequalities , Martingales , stopping times , uniform integrability , Wald’s equation

Rights: Copyright © 1997 Institute of Mathematical Statistics

Vol.25 • No. 4 • October 1997
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