Transportation-information inequalities for 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 Léonard in the i.i.d.\! case . Tensorization properties of $T_cI$ are also derived.
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  • HAL Id : hal-00158258, version 1
  • ARXIV : 0706.4193

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Arnaud Guillin, Christian Léonard, Liming Wu, Nian Yao. Transportation-information inequalities for Markov processes. Probability Theory and Related Fields, Springer Verlag, 2009, 144 (3-4), pp.669-695. ⟨hal-00158258⟩

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