The Annals of Probability

Moments of randomly stopped $U$-statistics

Tze Leung Lai and Victor H. de la Peña

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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.

Article information

Ann. Probab., Volume 25, Number 4 (1997), 2055-2081.

First available in Project Euclid: 7 June 2002

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60] 60F25: $L^p$-limit theorems
Secondary: 62L12: Sequential estimation

Decoupling inequalities martingales stopping times $U$-statistics Wald’s equation uniform integrability


de la Peña, Victor H.; Lai, Tze Leung. Moments of randomly stopped $U$-statistics. Ann. Probab. 25 (1997), no. 4, 2055--2081. doi:10.1214/aop/1023481120.

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