Exponential tail bounds and Large Deviation Principle for Heavy-Tailed U-Statistics

TL;DR

This paper establishes exponential tail bounds and a Large Deviation Principle for heavy-tailed U-statistics, revealing two decay regions and providing sharp constants for common cases, even with slower LDP speed than sample size.

Contribution

It introduces new exponential tail bounds and LDP results for heavy-tailed U-statistics, including sharp constants and two-region tail decay analysis.

Findings

Exponential upper bounds with Gaussian and kernel-like decay regions.

LDP with slower speed than sample size for heavy-tailed U-statistics.

Sharp constants and logarithmic limits for common U-statistics.

Abstract

We study deviation of U-statistics when samples have heavy-tailed distribution so the kernel of the U-statistic does not have bounded exponential moments at any positive point. We obtain an exponential upper bound for the tail of the U-statistics which clearly denotes two regions of tail decay, the first is a Gaussian decay and the second behaves like the tail of the kernel. For several common U-statistics, we also show the upper bound has the right rate of decay as well as sharp constants by obtaining rough logarithmic limits which in turn can be used to develop LDP for U-statistics. In spite of usual LDP results in the literature, processes we consider in this work have LDP speed slower than their sample size $n$.