December 2023 Efficient shape-constrained inference for the autocovariance sequence from a reversible Markov chain
Stephen Berg, Hyebin Song
Author Affiliations +
Ann. Statist. 51(6): 2440-2470 (December 2023). DOI: 10.1214/23-AOS2335


In this paper, we study the problem of estimating the autocovariance sequence resulting from a reversible Markov chain. A motivating application for studying this problem is the estimation of the asymptotic variance in central limit theorems for Markov chains. We propose a novel shape-constrained estimator of the autocovariance sequence, which is based on the key observation that the representability of the autocovariance sequence as a moment sequence imposes certain shape constraints. We examine the theoretical properties of the proposed estimator and provide strong consistency guarantees for our estimator. In particular, for geometrically ergodic reversible Markov chains, we show that our estimator is strongly consistent for the true autocovariance sequence with respect to an 2 distance, and that our estimator leads to strongly consistent estimates of the asymptotic variance. Finally, we perform empirical studies to illustrate the theoretical properties of the proposed estimator as well as to demonstrate the effectiveness of our estimator in comparison with other current state-of-the-art methods for Markov chain Monte Carlo variance estimation, including batch means, spectral variance estimators, and the initial convex sequence estimator.

Funding Statement

The authors gratefully acknowledge NSF support DMS-2311141.


Both authors contributed equally to this work. The authors would like to thank the anonymous referees and an Associate Editor for their constructive comments that improved the quality of this paper.


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Stephen Berg. Hyebin Song. "Efficient shape-constrained inference for the autocovariance sequence from a reversible Markov chain." Ann. Statist. 51 (6) 2440 - 2470, December 2023.


Received: 1 July 2022; Revised: 1 August 2023; Published: December 2023
First available in Project Euclid: 20 December 2023

MathSciNet: MR4682704
zbMATH: 07783622
Digital Object Identifier: 10.1214/23-AOS2335

Primary: 60J05 , 62G05
Secondary: 60J22

Keywords: asymptotic variance , autocovariance sequence estimation , Markov chain Monte Carlo , shape-constrained inference

Rights: Copyright © 2023 Institute of Mathematical Statistics


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Vol.51 • No. 6 • December 2023
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