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

TL;DR

This paper derives an exponential inequality for Hilbert-valued U-statistics of i.i.d. data, providing bounds that include exponential decay and tail terms, with applications to various types of U-statistics.

Contribution

It introduces a novel exponential inequality for Hilbert-valued U-statistics, extending existing bounds to more general kernels and statistical settings.

Findings

Established exponential bounds for Hilbert-valued U-statistics.

Extended inequalities to non-degenerate, weighted, and incomplete U-statistics.

Provided applications demonstrating the bounds' utility in different contexts.

Abstract

In this paper, we establish an exponential inequality for U-statistics of i.i.d. data, varying kernel and taking values in a separable Hilbert space. The bound are expressed as a sum of an exponential term plus an other one involving the tail of a sum of squared norms. We start by the degenerate case. Then we provide applications to U-statistics of not necessarily degenerate fixed kernel, weighted U-statistics and incomplete U-statistics.