Quantitative contraction rates for Markov chains on general state spaces

Andreas Eberle, Mateusz B. Majka

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Abstract

We investigate the problem of quantifying contraction coefficients of Markov transition kernels in Kantorovich (L1 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 Rd 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.

Original languageEnglish
Article number26
Number of pages36
JournalElectronic Journal of Probability
Volume24
DOIs
Publication statusPublished - 26 Mar 2019

Keywords

  • Couplings
  • Euler schemes
  • Markov chains
  • Metropolis algorithm
  • Quantitative bounds
  • Wasserstein distances

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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