The Annals of Applied Probability

Variance bounding Markov chains

Gareth O. Roberts and Jeffrey S. Rosenthal

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Abstract

We introduce a new property of Markov chains, called variance bounding. We prove that, for reversible chains at least, variance bounding is weaker than, but closely related to, geometric ergodicity. Furthermore, variance bounding is equivalent to the existence of usual central limit theorems for all L2 functionals. Also, variance bounding (unlike geometric ergodicity) is preserved under the Peskun order. We close with some applications to Metropolis–Hastings algorithms.

Article information

Source
Ann. Appl. Probab., Volume 18, Number 3 (2008), 1201-1214.

Dates
First available in Project Euclid: 26 May 2008

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1211819798

Digital Object Identifier
doi:10.1214/07-AAP486

Mathematical Reviews number (MathSciNet)
MR2418242

Zentralblatt MATH identifier
1142.60047

Subjects
Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Secondary: 65C40: Computational Markov chains 47A10: Spectrum, resolvent

Keywords
Markov chain Monte Carlo Metropolis–Hastings algorithm central limit theorem variance Peskun order geometric ergodicity spectrum

Citation

Roberts, Gareth O.; Rosenthal, Jeffrey S. Variance bounding Markov chains. Ann. Appl. Probab. 18 (2008), no. 3, 1201--1214. doi:10.1214/07-AAP486. https://projecteuclid.org/euclid.aoap/1211819798


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