Electronic Journal of Probability

Some universal estimates for reversible Markov chains

Mykhaylo Shkolnikov

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Reversible Markov chains convergence to equilibrium time of coupling entropy

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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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