Transport-information inequalities for Markov chains

This paper is the discrete time counterpart of the previous work in the continuous time case by Guillin, Léonard, the second named author and Yao [Probab. Theory Related Fields 144 (2009), 669.695]. We investigate the following transport-information T.I inequality: .(T.(.,.)).I(.|P,.) for all probability measures . on some metric space (.,d), where . is an invariant and ergodic probability measure of some given transition kernel P(x,dy) T.(.,.) is some transportation cost from . , I(.|P,.) is the Donsker.Varadhan information of . with respect to (P,.) and .:[0,.).[0,.] is some left continuous increasing function. Using large deviation techniques, we show that T.I is equivalent to some concentration inequality for the occupation measure of the .-reversible Markov chain (Xn)n.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.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(.|P,.) has no closed expression, contrary to the continuous time or independent and identically distributed case.