Theory of Probability &Its Applications, Volume 66, Issue 3, Page 408-429, January 2021.
We establish an exponential inequality for degenerated $U$-statistics of order $r$ of independent and identically distributed (i.i.d.) data. This inequality gives a control of the tail of the maxima absolute values of the $U$-statistic by the sum of the two terms: an exponential term and one involving the tail of $h(X_1,\dots,X_r)$. We also give a version for not necessarily degenerated $U$-statistics having a symmetric kernel and furnish an application to the convergence rates in the Marcinkiewicz law of large numbers. Application to the invariance principle in Hölder spaces is also considered.

中文翻译：

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IID 数据的 $U$-统计的指数不等式

Theory of Probability & Its Applications，第 66 卷，第 3 期，第 408-429 页，2021 年 1 月。
我们为独立同分布 (iid) 数据的 $r$ 阶退化 $U$ 统计量建立指数不等式。这种不等式通过两项之和来控制 $U$ 统计量的最大值绝对值的尾部：指数项和涉及 $h(X_1,\dots,X_r)$ 尾部的一项。我们还给出了具有对称核的不一定退化的 $U$ 统计量的版本，并提供了适用于 Marcinkiewicz 大数定律中的收敛率的应用程序。还考虑了 Hölder 空间中不变性原理的应用。