A simple variance inequality for U-statistics of a Markov chain with applications

TL;DR

This paper derives an explicit variance inequality for U-statistics based on ergodic Markov chains, enabling near-optimal strong law of large numbers under computable mixing conditions.

Contribution

It introduces a simple, explicit variance inequality for U-statistics of ergodic Markov chains with practical bounds on mixing rates.

Findings

Established a variance inequality with explicit constants

Derived a strong law of large numbers for Markov chain U-statistics

Applied the inequality to near-optimal convergence conditions

Abstract

We establish a simple variance inequality for U-statistics whose underlying sequence of random variables is an ergodic Markov Chain. The constants in this inequality are explicit and depend on computable bounds on the mixing rate of the Markov Chain. We apply this result to derive the strong law of large number for U-statistics of a Markov Chain under conditions which are close from being optimal.