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A new proof of convergence of MCMC via the ergodic theorem

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  • Asmussen, Søren
  • Glynn, Peter W.

Abstract

A key result underlying the theory of MCMC is that any [eta]-irreducible Markov chain having a transition density with respect to [eta] and possessing a stationary distribution [pi] is automatically positive Harris recurrent. This paper provides a short self-contained proof of this fact using the ergodic theorem in its standard form as the most advanced tool.

Suggested Citation

  • Asmussen, Søren & Glynn, Peter W., 2011. "A new proof of convergence of MCMC via the ergodic theorem," Statistics & Probability Letters, Elsevier, vol. 81(10), pages 1482-1485, October.
    • Handle: RePEc:eee:stapro:v:81:y:2011:i:10:p:1482-1485
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    Cited by:

    1. Ankush Agarwal & Stefano de Marco & Emmanuel Gobet & Gang Liu, 2017. "Rare event simulation related to financial risks: efficient estimation and sensitivity analysis," Working Papers hal-01219616, HAL.
    2. Khare, Kshitij & Hobert, James P., 2012. "Geometric ergodicity of the Gibbs sampler for Bayesian quantile regression," Journal of Multivariate Analysis, Elsevier, vol. 112(C), pages 108-116.
    3. Rudolf, Daniel & Schweizer, Nikolaus, 2015. "Error bounds of MCMC for functions with unbounded stationary variance," Statistics & Probability Letters, Elsevier, vol. 99(C), pages 6-12.
    4. Bryant Davis & James P. Hobert, 2021. "On the Convergence Complexity of Gibbs Samplers for a Family of Simple Bayesian Random Effects Models," Methodology and Computing in Applied Probability, Springer, vol. 23(4), pages 1323-1351, December.
    5. Ye Chen & Ilya O. Ryzhov, 2020. "Technical Note—Consistency Analysis of Sequential Learning Under Approximate Bayesian Inference," Operations Research, INFORMS, vol. 68(1), pages 295-307, January.
    6. Kshitij Khare & Malay Ghosh, 2022. "MCMC Convergence for Global-Local Shrinkage Priors," Journal of Quantitative Economics, Springer;The Indian Econometric Society (TIES), vol. 20(1), pages 211-234, September.
    7. Halme, Merja & Kallio, Markku, 2014. "Likelihood estimation of consumer preferences in choice-based conjoint analysis," European Journal of Operational Research, Elsevier, vol. 239(2), pages 556-564.

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