Deviation and moment inequalities for Banach-valued $U$-statistics

TL;DR

This paper establishes deviation inequalities and moment bounds for Banach-valued U-statistics, with applications to laws of large numbers and functional central limit theorems in Banach spaces.

Contribution

It introduces new deviation and moment inequalities for Banach-valued U-statistics under smoothness conditions, extending classical results to Banach space settings.

Findings

Deviation inequality for Banach-valued U-statistics

Rates in the law of large numbers for U-statistics

A H{"o}lderian functional central limit theorem

Abstract

We show a deviation inequality for U-statistics of independent data taking values in a separable Banach space which satisfies some smoothness assumptions. We then provide applications to rates in the law of large numbers for U-statistics, a H{\"o}lderian functional central limit theorem and a moment inequality for incomplete $U$-statistics.