The Annals of Probability

Sub-Bernoulli Functions, Moment Inequalities and Strong Laws for Nonnegative and symmetrized U-Statistics

Cun-Hui Zhang

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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.

Article information

Ann. Probab., Volume 27, Number 1 (1999), 432-453.

First available in Project Euclid: 29 May 2002

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F15: Strong theorems
Secondary: 60G50: Sums of independent random variables; random walks

Sub-Bernoulli function strong law of large numbers moment inequality exponential inequality tail probability $U$-statistics


Zhang, Cun-Hui. Sub-Bernoulli Functions, Moment Inequalities and Strong Laws for Nonnegative and symmetrized U -Statistics. Ann. Probab. 27 (1999), no. 1, 432--453. doi:10.1214/aop/1022677268.

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