Open Access
2013 Some universal estimates for reversible Markov chains
Mykhaylo Shkolnikov
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Electron. J. Probab. 18: 1-17 (2013). DOI: 10.1214/EJP.v18-1749


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 />


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Mykhaylo Shkolnikov. "Some universal estimates for reversible Markov chains." Electron. J. Probab. 18 1 - 17, 2013.


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

zbMATH: 06247180
MathSciNet: MR3035739
Digital Object Identifier: 10.1214/EJP.v18-1749

Primary: 60J10
Secondary: 94A17

Keywords: Convergence to equilibrium , Entropy , reversible Markov chains , time of coupling

Vol.18 • 2013
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