## The Annals of Probability

### Moments of randomly stopped $U$-statistics

#### Abstract

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

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

Dates
First available in Project Euclid: 7 June 2002

https://projecteuclid.org/euclid.aop/1023481120

Digital Object Identifier
doi:10.1214/aop/1023481120

Mathematical Reviews number (MathSciNet)
MR1487445

Zentralblatt MATH identifier
0902.60037

#### Citation

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. https://projecteuclid.org/euclid.aop/1023481120

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• NEW YORK, NEW YORK 10027 STANFORD, CALIFORNIA 94305 E-MAIL: karola@stat.stanford.edu