Transportation-information inequalities for Markov processes

Abstract

In this paper, one investigates the transportation-information T c I inequalities: α(T c (ν, μ)) ≤ I (ν|μ) for all probability measures ν on a metric space Xd, where μ is a given probability measure, T c (ν, μ) is the transportation cost from ν to μ with respect to the cost function c(x, y) on X2 , I(ν|μ) is the Fisher-Donsker-Varadhan information of ν with respect to μ and α : [0, ∞) → [0, ∞] is a left continuous increasing function. Using large deviation techniques, it is shown that T c I is equivalent to some concentration inequality for the occupation measure of a μ-reversible ergodic Markov process related to I(•|μ). The tensorization property of T c I and comparisons of T c I with Poincaré and log-Sobolev inequalities are investigated. Several easy-to-check sufficient conditions are provided for special important cases of T c I and several examples are worked out. © 2008 Springer-Verlag.