Uniform Limit Theorem and tail estimates for parametric u-statistics

TL;DR

This paper establishes conditions for weak convergence of u-statistics in function spaces and provides non-asymptotic tail estimates using martingale representations and metric entropy.

Contribution

It introduces new sufficient conditions for weak convergence and derives non-asymptotic tail bounds for the uniform deviation of u-statistics.

Findings

Weak convergence criteria for u-statistics in function spaces.

Non-asymptotic tail estimates for the uniform norm of u-statistics.

Use of martingale representation and metric entropy in tail analysis.

Abstract

We deduce in this paper the sufficient conditions for weak convergence of centered and normed deviation of the u-statistics with values in the space of the real valued continuous function defined on some compact metric space. We obtain also a non-asymptotic and non-improvable up to multiplicative constant moment and exponential tail estimates for distribution for the uniform norm of centered and naturally normed deviation of u-statistics by means of its martingale representation. Our results are formulated in a very popular and natural terms of metric entropy in the distance (distances) generated by the introduced random processes (fields).