Sub-Bernoulli functions, moment inequalities and strong laws for nonnegative and symmetrized U-statistics

• 成果类型: Article
• 署名作者: Zhang, CH
• 署名单位: Rutgers University System; Rutgers University New Brunswick
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/aop/1022677268
• 发表日期: 1999
• 页码: 432-453
• 关键词: large numbers; PRODUCTS; forms; sums

摘要

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.