A characterization of dimension free concentration in terms of transportation inequalities

Nathael Gozlan (1)

1 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées

The aim of this paper is to show that a probability measure concentrates independently of the dimension like a gaussian measure if and only if it verifies Talagrand's $\T_2$ transportation-cost inequality. This theorem permits us to give a new and very short proof of a result of Otto and Villani. Generalizations to other types of concentration are also considered. In particular, one shows that the Poincaré inequality is equivalent to a certain form of dimension free exponential concentration. The proofs of these results rely on simple Large Deviations techniques.

Mots clés: Concentration of measure; Transportation-cost inequalities; Sanov's Theorem; Logarithmic-Sobolev inequalities

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

Nathael Gozlan. A characterization of dimension free concentration in terms of transportation inequalities. Annals of Probability, 2009, 37 (6), pp.2480--2498. ⟨10.1214/09-AOP470⟩. ⟨hal-00274548v1⟩