Transportation-information inequalities for Markov processes

Abstract

In this paper, one investigates the following type of transportation-information $T_cI$ inequalities: $α(T_c(ν,μ))\le I(ν|μ)$ for all probability measures $ν$ on some metric space $(\XX, d)$, where $μ$ is a given probability measure, $T_c(ν,μ)$ is the transportation cost from $ν$ to $μ$ with respect to some cost function $c(x,y)$ on $\XX^2$, $I(ν|μ)$ is the Fisher-Donsker-Varadhan information of $ν$ with respect to $μ$ and $α: [0,\infty)\to [0,\infty]$ is some left continuous increasing function. Using large deviation techniques, it is shown that $T_cI$ is equivalent to some concentration inequality for the occupation measure of a $μ$-reversible ergodic Markov process related to $I(\cdot|μ)$, a counterpart of the characterizations of transportation-entropy inequalities, recently obtained by Gozlan and Léonard in the i.i.d. case . Tensorization properties of $T_cI$ are also derived.