Stability under mixtures of transportation inequalities and restricted log-Sobolev inequalities

Abstract

We prove results about stability under mixtures for transport--entropy inequalities and restricted functional inequalities of Poincaré and log-Sobolev type. We treat Talagrand inequalities of type ${\mathbf{T}}_2$ and ${\mathbf{T}}_1$ together with their generalizations, as well as weak transportation inequalities introduced by Gozlan et al. Our main results provide mild sufficient conditions under which a given inequality may be transferred from mixture components to the mixture itself.