Transportation-information inequalities for Markov processes

TL;DR

This paper explores transportation-information inequalities for Markov processes, establishing their equivalence to concentration inequalities and analyzing their tensorization properties, extending recent i.i.d. case results to Markov settings.

Contribution

It introduces transportation-information inequalities for Markov processes, linking them to concentration inequalities and tensorization, thus extending existing i.i.d. results to Markov processes.

Findings

$T_cI$ inequalities are equivalent to concentration inequalities for occupation measures.

Tensorization properties of $T_cI$ are established.

The work extends transportation-entropy inequality characterizations to Markov processes.

Abstract

In this paper, one investigates the following type of transportation-information $T_cI$ inequalities: $\alpha(T_c(\nu,\mu))\le I(\nu|\mu)$ for all probability measures $\nu$ on some metric space $(\XX, d)$, where $\mu$ is a given probability measure, $T_c(\nu,\mu)$ is the transportation cost from $\nu$ to $\mu$ with respect to some cost function $c(x,y)$ on $\XX^2$, $I(\nu|\mu)$ is the Fisher-Donsker-Varadhan information of $\nu$ with respect to $\mu$ and $\alpha: [0,\infty)\to [0,\infty]$ 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 $\mu$-reversible ergodic Markov process related to $I(\cdot|\mu)$, a counterpart of the characterizations of transportation-entropy inequalities, recently obtained by Gozlan and L\'eonard in the i.i.d. case . Tensorization properties of $T_cI$ are also derived.