Transport-information inequalities for Markov chains

Abstract

This paper is the discrete time counterpart of the previous work in the continuous time case by Guillin, Léonard, the second named author and Yao [Probab. Theory Related Fields 144 (2009), 669–695]. We investigate the following transport-information $T_{\mathcal{V}}I$ inequality: $\alpha (T_{\mathcal{V}}(\nu ,\mu ))\le I(\nu |P,\mu )$ for all probability measures $\nu $ on some metric space $(\mathcal{X},d)$, where $\mu $ is an invariant and ergodic probability measure of some given transition kernel $P(x,dy)$, $T_{\mathcal{V}}(\nu ,\mu )$ is some transportation cost from $\nu $ to $\mu $, $I(\nu |P,\mu )$ is the Donsker–Varadhan information of $\nu $ with respect to $(P,\mu )$ and $\alpha :[0,\infty )\to [0,\infty ]$ is some left continuous increasing function. Using large deviation techniques, we show that $T_{\mathcal{V}}I$ is equivalent to some concentration inequality for the occupation measure of the $\mu $-reversible Markov chain $(X_{n})_{n\ge 0}$ with transition probability $P(x,dy)$. Its relationships with the transport-entropy inequalities are discussed. Several easy-to-check sufficient conditions are provided for $T_{\mathcal{V}}I$. We show the usefulness and sharpness of our general results by a number of applications and examples. The main difficulty resides in the fact that the information $I(\nu |P,\mu )$ has no closed expression, contrary to the continuous time or independent and identically distributed case.