Concentration inequalities for Markov chains by Marton couplings and spectral methods
Abstract
We prove a version of McDiarmid's bounded differences inequality for Markov chains, with constants proportional to the mixing time of the chain. We also show variance bounds and Bernstein-type inequalities for empirical averages of Markov chains. In the case of non-reversible chains, we introduce a new quantity called the "pseudo spectral gap", and show that it plays a similar role for non-reversible chains as the spectral gap plays for reversible chains. Our techniques for proving these results are based on a coupling construction of Katalin Marton, and on spectral techniques due to Pascal Lezaud. The pseudo spectral gap generalises the multiplicative reversiblication approach of Jim Fill.
Keywords
Cite
@article{arxiv.1212.2015,
title = {Concentration inequalities for Markov chains by Marton couplings and spectral methods},
author = {Daniel Paulin},
journal= {arXiv preprint arXiv:1212.2015},
year = {2018}
}
Comments
42 pages. In the previous version, the proofs of Bernstein's inequalities for Markov chains on general state spaces were using an argument from the proofs of Theorems 1.1 and 1.5 on pages 100-101 of the doctoral thesis of Pascal Lezaud. A part of that argument was incomplete. In this version, we correct this