Weighted Csiszár-Kullback-Pinsker inequalities and applications to transportation inequalities

Abstract

We strengthen the usual Csisz´ ar-Kullback-Pinsker in- equality by allowing weights in the total variation norm; admissible weights depend on the decay of the reference probability measure.We use this re- sult to derive transportation inequalities involving Wasserstein distances for various exponents: in particular, we recover the equivalence between a T1 inequality and the existence of a square-exponential moment. Then we give a variant of the results obtained by Djellout, Guillin and Wu [5] about transportation inequalities for random dynamical systems, in which a sufficient condition is expressed in terms of exponential moments. An unpublished result by Blower [1] about the perturbation of a T2 inequality is also recovered and generalized.