Electronic Journal of Probability

Quantitative contraction rates for Markov chains on general state spaces

Andreas Eberle and Mateusz B. Majka

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We investigate the problem of quantifying contraction coefficients of Markov transition kernels in Kantorovich ($L^1$ Wasserstein) distances. For diffusion processes, relatively precise quantitative bounds on contraction rates have recently been derived by combining appropriate couplings with carefully designed Kantorovich distances. In this paper, we partially carry over this approach from diffusions to Markov chains. We derive quantitative lower bounds on contraction rates for Markov chains on general state spaces that are powerful if the dynamics is dominated by small local moves. For Markov chains on $\mathbb R^d$ with isotropic transition kernels, the general bounds can be used efficiently together with a coupling that combines maximal and reflection coupling. The results are applied to Euler discretizations of stochastic differential equations with non-globally contractive drifts, and to the Metropolis adjusted Langevin algorithm for sampling from a class of probability measures on high dimensional state spaces that are not globally log-concave.

Article information

Electron. J. Probab., Volume 24 (2019), paper no. 26, 36 pp.

Received: 21 August 2018
Accepted: 2 March 2019
First available in Project Euclid: 26 March 2019

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Primary: 60J05: Discrete-time Markov processes on general state spaces 60J22: Computational methods in Markov chains [See also 65C40] 65C05: Monte Carlo methods 65C30: Stochastic differential and integral equations 65C40: Computational Markov chains

Markov chains Wasserstein distances quantitative bounds couplings Euler schemes Metropolis algorithm

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Eberle, Andreas; Majka, Mateusz B. Quantitative contraction rates for Markov chains on general state spaces. Electron. J. Probab. 24 (2019), paper no. 26, 36 pp. doi:10.1214/19-EJP287. https://projecteuclid.org/euclid.ejp/1553565777

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