In this paper, we prove exponential tail bounds for canonical (or degenerate) $U$-statistics and $U$-processes under exponential-type tail assumptions on the kernels. Most of the existing results in the relevant literature often assume bounded kernels or obtain sub-optimal tail behavior under unbounded kernels. We obtain sharp rates and optimal tail behavior under sub-Weibull kernel functions. Some examples from nonparametric and semiparametric statistics literature are considered.

翻译：本文在核函数满足指数型尾部假设的条件下，证明了典型（或退化）$U$-统计量与$U$-过程的指数型尾部界。相关文献中的现有结果通常假设核函数有界，或在无界核条件下得到次优的尾部行为。我们在次Weibull核函数假设下获得了尖锐的收敛速率与最优尾部行为。文中考虑了来自非参数与半参数统计文献中的若干示例。