The Annals of Statistics

Kernel estimators of asymptotic variance for adaptive Markov chain Monte Carlo

Yves F. Atchadé

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We study the asymptotic behavior of kernel estimators of asymptotic variances (or long-run variances) for a class of adaptive Markov chains. The convergence is studied both in Lp and almost surely. The results also apply to Markov chains and improve on the existing literature by imposing weaker conditions. We illustrate the results with applications to the GARCH(1, 1) Markov model and to an adaptive MCMC algorithm for Bayesian logistic regression.

Article information

Ann. Statist., Volume 39, Number 2 (2011), 990-1011.

First available in Project Euclid: 8 April 2011

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Zentralblatt MATH identifier

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 60C05: Combinatorial probability

Adaptive Markov chain Monte Carlo kernel estimators of asymptotic variance


Atchadé, Yves F. Kernel estimators of asymptotic variance for adaptive Markov chain Monte Carlo. Ann. Statist. 39 (2011), no. 2, 990--1011. doi:10.1214/10-AOS828.

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Supplemental materials

  • Supplementary material: Supplement to “Kernel estimators of asymptotic variance for adaptive Markov chain Monte Carlo”. The proofs of Theorems 4.1–4.3 require some technical and lengthy arguments that we develop in this supplement.