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August 2021 On the limitations of single-step drift and minorization in Markov chain convergence analysis
Qian Qin, James P. Hobert
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Ann. Appl. Probab. 31(4): 1633-1659 (August 2021). DOI: 10.1214/20-AAP1628

Abstract

Over the last three decades, there has been a considerable effort within the applied probability community to develop techniques for bounding the convergence rates of general state space Markov chains. Most of these results assume the existence of drift and minorization (d&m) conditions. It has often been observed that convergence rate bounds based on single-step d&m tend to be overly conservative, especially in high-dimensional situations. This article builds a framework for studying this phenomenon. It is shown that any convergence rate bound based on a set of d&m conditions cannot do better than a certain unknown optimal bound. Strategies are designed to put bounds on the optimal bound itself, and this allows one to quantify the extent to which a d&m-based convergence rate bound can be sharp. The new theory is applied to several examples, including a Gaussian autoregressive process (whose true convergence rate is known), and a Metropolis adjusted Langevin algorithm. The results strongly suggest that convergence rate bounds based on single-step d&m conditions are quite inadequate in high-dimensional settings.

Funding Statement

The second author was supported by NSF Grant DMS-15-11945.

Acknowledgments

We thank the Editor and four anonymous reviewers for helpful comments and suggestions.

Citation

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Qian Qin. James P. Hobert. "On the limitations of single-step drift and minorization in Markov chain convergence analysis." Ann. Appl. Probab. 31 (4) 1633 - 1659, August 2021. https://doi.org/10.1214/20-AAP1628

Information

Received: 1 March 2020; Revised: 1 September 2020; Published: August 2021
First available in Project Euclid: 15 September 2021

Digital Object Identifier: 10.1214/20-AAP1628

Subjects:
Primary: 60J05

Keywords: convergence rate , coupling , geometric ergodicity , high-dimensional inference , optimal bound , quantitative bound , renewal theory

Rights: Copyright © 2021 Institute of Mathematical Statistics

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Vol.31 • No. 4 • August 2021
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