A criterion for Talagrand's quadratic transportation cost inequality

Patrick Cattiaux (1, 2) , Arnaud Guillin (1)

1 MODAL'X - Modélisation aléatoire de Paris X
2 CMAP - Centre de Mathématiques Appliquées

We show that the quadratic transportation cost inequality $T_2$ is equivalent to both a Poincaré inequality and a strong form of the Gaussian concentration property. The main ingredient in the proof is a new family of inequalities, called modified quadratic transportation cost inequalities in the spirit of the modified logarithmic-Sobolev inequalities by Bobkov and Ledoux \cite{BL97}, that are shown to hold as soon as a Poincaré inequality is satisfied.

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

Patrick Cattiaux, Arnaud Guillin. A criterion for Talagrand's quadratic transportation cost inequality. 2003. ⟨hal-00133642⟩