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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Abstract

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

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

Dates
First available in Project Euclid: 29 May 2002

Permanent link to this document
https://projecteuclid.org/euclid.aop/1022677268

Digital Object Identifier
doi:10.1214/aop/1022677268

Mathematical Reviews number (MathSciNet)
MR1681165

Zentralblatt MATH identifier
0951.60028

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

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

Citation

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. https://projecteuclid.org/euclid.aop/1022677268


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