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August 2010 Quantitative bounds for Markov chain convergence: Wasserstein and total variation distances
Neal Madras, Deniz Sezer
Bernoulli 16(3): 882-908 (August 2010). DOI: 10.3150/09-BEJ238


We present a framework for obtaining explicit bounds on the rate of convergence to equilibrium of a Markov chain on a general state space, with respect to both total variation and Wasserstein distances. For Wasserstein bounds, our main tool is Steinsaltz’s convergence theorem for locally contractive random dynamical systems. We describe practical methods for finding Steinsaltz’s “drift functions” that prove local contractivity. We then use the idea of “one-shot coupling” to derive criteria that give bounds for total variation distances in terms of Wasserstein distances. Our methods are applied to two examples: a two-component Gibbs sampler for the Normal distribution and a random logistic dynamical system.


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Neal Madras. Deniz Sezer. "Quantitative bounds for Markov chain convergence: Wasserstein and total variation distances." Bernoulli 16 (3) 882 - 908, August 2010.


Published: August 2010
First available in Project Euclid: 6 August 2010

zbMATH: 1284.60143
MathSciNet: MR2730652
Digital Object Identifier: 10.3150/09-BEJ238

Keywords: convergence rate , coupling , Gibbs sampler , iterated random functions , local contractivity , logistic map , Markov chain , Random dynamical system , total variation distance , Wasserstein distance

Rights: Copyright © 2010 Bernoulli Society for Mathematical Statistics and Probability


Vol.16 • No. 3 • August 2010
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