Electronic Journal of Probability

Some universal estimates for reversible Markov chains

Mykhaylo Shkolnikov

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Abstract

We obtain universal estimates on the convergence to equilibrium and the times of coupling for continuous time irreducible reversible finite-state Markov chains, both in the total variation and in the $L^2$ norms. The estimates in total variation norm are obtained using a novel identity relating the convergence to equilibrium of a reversible Markov chain to the increase in the entropy of its one-dimensional distributions. In addition, we propose a universal way of defining the ultrametric partition structure on the state space of such Markov chains. Finally, for chains reversible with respect to the uniform measure, we show how the global convergence to equilibrium can be controlled using the entropy accumulated by the chain. <br />

Article information

Source
Electron. J. Probab., Volume 18 (2013), paper no. 11, 17 pp.

Dates
Accepted: 17 January 2013
First available in Project Euclid: 4 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1465064236

Digital Object Identifier
doi:10.1214/EJP.v18-1749

Mathematical Reviews number (MathSciNet)
MR3035739

Zentralblatt MATH identifier
06247180

Subjects
Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Secondary: 94A17: Measures of information, entropy

Keywords
Reversible Markov chains convergence to equilibrium time of coupling entropy

Rights
This work is licensed under a Creative Commons Attribution 3.0 License.

Citation

Shkolnikov, Mykhaylo. Some universal estimates for reversible Markov chains. Electron. J. Probab. 18 (2013), paper no. 11, 17 pp. doi:10.1214/EJP.v18-1749. https://projecteuclid.org/euclid.ejp/1465064236


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References

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