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

Eric Moulines, Alexey Naumov, Sergey Samsonov

TL;DR

This work analyzes concentration properties of quadratic forms arising from Markov chains and, in particular, of the overlapped batch mean (OBM) estimator for the chain's asymptotic variance under uniform geometric ergodicity. It employs a martingale-decomposition framework to derive explicit $p$-th moment bounds with sharp dependence on the moment order and the chain's mixing time, and it translates these results into finite-sample concentration bounds for the OBM estimator. The paper provides a detailed remainder decomposition and leverages Burkholder and Rosenthal-type inequalities alongside probabilistic inequalities to control estimation error. The findings yield concrete guidance on variance estimation in MCMC, including how the sample size $n$ and batch size $b_n$ interact with the mixing time to influence concentration.

Abstract

In this paper, we study the concentration properties of quadratic forms associated with Markov chains using the martingale decomposition method introduced by Atchadé 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.

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

TL;DR

This work analyzes concentration properties of quadratic forms arising from Markov chains and, in particular, of the overlapped batch mean (OBM) estimator for the chain's asymptotic variance under uniform geometric ergodicity. It employs a martingale-decomposition framework to derive explicit -th moment bounds with sharp dependence on the moment order and the chain's mixing time, and it translates these results into finite-sample concentration bounds for the OBM estimator. The paper provides a detailed remainder decomposition and leverages Burkholder and Rosenthal-type inequalities alongside probabilistic inequalities to control estimation error. The findings yield concrete guidance on variance estimation in MCMC, including how the sample size and batch size interact with the mixing time to influence concentration.

Abstract

In this paper, we study the concentration properties of quadratic forms associated with Markov chains using the martingale decomposition method introduced by Atchadé 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 -th moment of the difference between the OBM estimator and the asymptotic variance of the Markov chain with explicit dependence upon and mixing time of the underlying Markov chain.
Paper Structure (7 sections, 9 theorems, 105 equations)

This paper contains 7 sections, 9 theorems, 105 equations.

Table of Contents

  1. Introduction
  2. Problem setting.
  3. Applications to the concentration properties of the OBM estimator
  4. Postponed proofs
  5. Proof of \ref{['prop:quadr_form_decomposition']}
  6. Proof of the error representation from \ref{['lem:bar_r_n_representation']}
  7. Probability inequalities

Key Result

Proposition 1

Assume assum:UGE. Then for any bounded function $f: \mathsf{Z} \to \mathbb{R}$, it holds that where we have defined

Theorems & Definitions (17)

  • Proposition 1: Lemma 2.2 in atchade2014martingale
  • proof
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Theorem 1
  • proof
  • Lemma 3
  • proof
  • ...and 7 more