A note on concentration inequalities for the overlapped batch mean variance estimators for Markov chains

TL;DR

This paper derives concentration inequalities for overlapped batch mean estimators of asymptotic variance in geometrically ergodic Markov chains, providing explicit bounds based on moments and mixing times.

Contribution

It introduces new concentration inequalities for OBM estimators in Markov chains, with explicit dependence on moments and mixing properties.

Findings

Explicit p-th moment bounds for OBM estimators

Dependence of bounds on mixing time and chain properties

Improved understanding of variance estimator concentration

Abstract

In this paper, we study the concentration properties of quadratic forms associated with Markov chains using the martingale decomposition method introduced by Atchad\'e and Cattaneo (2014). In particular, we derive concentration inequalities for the overlapped batch mean (OBM) estimators of the asymptotic variance for uniformly geometrically ergodic Markov chains. Our main result provides an explicit control of the $p$-th moment of the difference between the OBM estimator and the asymptotic variance of the Markov chain with explicit dependence upon $p$ and mixing time of the underlying Markov chain.