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January 1999 Sub-Bernoulli Functions, Moment Inequalities and Strong Laws for Nonnegative and symmetrized U-Statistics
Cun-Hui Zhang
Ann. Probab. 27(1): 432-453 (January 1999). DOI: 10.1214/aop/1022677268


This paper concerns moment and tail probability inequalities and the strong law of large numbers for $U$-statistics with nonnegative or symmetrized kernels and their multisample and decoupled versions. Sub-Bernoulli functions are used to obtain the moment and tail probability inequalities, which are then used to obtain necessary and sufficient conditions for the almost sure convergence to zero of normalized $U$-statistics with nonnegative or completely symmetrized kernels, without further regularity conditions on the kernel or the distribution of the population, for normalizing constants satisfying a simple condition. Moments of $U$-statistics are bounded from above and below by that of maxima of certain kernels, up to scaling constants. The multisample and decoupled versions of these results are also considered.


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Cun-Hui Zhang. "Sub-Bernoulli Functions, Moment Inequalities and Strong Laws for Nonnegative and symmetrized U-Statistics." Ann. Probab. 27 (1) 432 - 453, January 1999.


Published: January 1999
First available in Project Euclid: 29 May 2002

zbMATH: 0951.60028
MathSciNet: MR1681165
Digital Object Identifier: 10.1214/aop/1022677268

Primary: 60F15
Secondary: 60G50

Keywords: $U$-statistics , Exponential inequality , moment inequality , Strong law of large numbers , Sub-Bernoulli function , tail probability

Rights: Copyright © 1999 Institute of Mathematical Statistics


Vol.27 • No. 1 • January 1999
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