Concentration inequality for U-statistics of order two for uniformly ergodic Markov chains

TL;DR

This paper establishes a new concentration inequality for U-statistics of order two in uniformly ergodic Markov chains, extending previous results for independent variables to dependent Markov processes.

Contribution

It introduces a concentration inequality that accounts for dependence in Markov chains with kernels that depend on indices, using martingale and ergodic techniques.

Findings

Recovers the convergence rate of independent case for Markov chains

Allows dependence of kernels on indices, unlike standard methods

Provides sharper bounds when starting from the invariant distribution

Abstract

We prove a new concentration inequality for U-statistics of order two for uniformly ergodic Markov chains. Working with bounded and $\pi$-canonical kernels, we show that we can recover the convergence rate of Arcones and Gin{\'e} who proved a concentration result for U-statistics of independent random variables and canonical kernels. Our result allows for a dependence of the kernels $h_{i,j}$ with the indexes in the sums, which prevents the use of standard blocking tools. Our proof relies on an inductive analysis where we use martingale techniques, uniform ergodicity, Nummelin splitting and Bernstein's type inequality. Assuming further that the Markov chain starts from its invariant distribution, we prove a Bernstein-type concentration inequality that provides sharper convergence rate for small variance terms.