An exponential inequality for $U$-statistics of i.i.d. data

TL;DR

This paper derives an exponential inequality for $U$-statistics of i.i.d. data, providing bounds on their tail behavior and applications to convergence rates and invariance principles.

Contribution

It introduces a new exponential inequality for degenerated and non-degenerated $U$-statistics with symmetric kernels, extending tail control methods.

Findings

Provides tail bounds for $U$-statistics of arbitrary order.

Establishes convergence rates in the Marcinkiewicz law of large numbers.

Applies to invariance principles in H"older spaces.

Abstract

We establish an exponential inequality for degenerated $U$-statistics of order $r$ of 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 two terms: an exponential term and one involving the tail of $h\left(X_1,\dots,X_r\right)$. 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 invariance principle in H\"older spaces is also considered.