On fine properties of mixtures with respect to concentration of measure and Sobolev type inequalities

Djalil Chafai (1, 2) , Florent Malrieu (3)

1 IMT - Institut de Mathématiques de Toulouse UMR5219
2 UPTE - Physiopathologie et Toxicologie Expérimentales
3 IRMAR - Institut de Recherche Mathématique de Rennes

Mixtures are convex combinations of laws. Despite this simple definition, a mixture can be far more subtle than its mixed components. For instance, mixing Gaussian laws may produce a wild potential with multiple wells. We study in the present work fine properties of mixtures with respect to concentration of measure and Gross type functional inequalities. We provide sharp Laplace bounds for Lipschitz functions in the case of generic mixtures, involving a transportation cost diameter of the mixed family. We also provide precise upper bounds for two-components mixtures. Additionally, our analysis of Gross type inequalities for two-components mixtures reveals natural relations with some kind of band isoperimetry and support constrained interpolation via mass transportation. We show that the Poincaré constant of a two-components mixture may remain bounded as the mixture proportion goes to 0 or 1 while the Gross constant may surprisingly blow up. Additionally, this counter-intuitive result is not reducible to support disconnections. As far as mixture of distributions are concerned, the Gross inequality is less stable than the sub-Gaussian concentration for Lipschitz functions. We illustrate our results on a gallery of concrete two-components mixtures.

Mots clés: Mixtures of distributions; finite Gaussian mixtures; functional inequalities; concentration of measure; transportation of measure; Sobolev type inequalities; Gaussian bounds; tails probabilities; deviation inequalities.; deviation inequalities

Domaines: Probabilités [math.PR]; Analyse fonctionnelle [math.FA]

Djalil Chafai, Florent Malrieu. On fine properties of mixtures with respect to concentration of measure and Sobolev type inequalities. 2008. ⟨hal-00277936v1⟩