Transport-information inequalities for Markov chains

This paper is the discrete time counterpart of the previous work in the continuous time case by Guillin, Leonard, 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.