Concentration of measure inequalities for Markov chains and $\Phi$-mixing processes

Abstract

We prove concentration inequalities for some classes of Markov chains and $\Phi$-mixing processes, with constants independent of the size of the sample, that extend the inequalities for product measures of Talagrand. The method is based on information inequalities put forwardby Marton in case of contracting Markov chains. Using a simple duality argument on entropy, our results also include the family of logarithmic Sobolev inequalities for convex functions. Applications to bounds on supremum of dependent empirical processes complete this work.