Poincaré inequalities and dimension free concentration of measure

Gozlan, Nathael

Annales de l'I.H.P. Probabilités et statistiques, Tome 46 (2010) no. 3, pp. 708-739.

Dans cet article, nous introduisons des inégalités de Poincaré pour des métriques non-euclidiennes sur ℝd et nous montrons qu'elles entraînent des inégalités de concentrations adimensionnelles pour les mesures produits. Cette technique nous permet d'atteindre un spectre très large de taux de concentration, aussi bien sous et sur-gaussiens. Par ailleurs, nous montrons que ces inégalités de Poincaré admettent des formes fonctionnelles équivalentes en termes d'inégalités de transport et d'inf-convolution. Enfin, nous donnons des conditions suffisantes pour ces inégalités de Poincaré et nous les comparons aux inégalités super-Poincaré.

In this paper, we consider Poincaré inequalities for non-euclidean metrics on ℝd. These inequalities enable us to derive precise dimension free concentration inequalities for product measures. This technique is appropriate for a large scope of concentration rate: between exponential and gaussian and beyond. We give equivalent functional forms of these Poincaré type inequalities in terms of transportation-cost inequalities and inf-convolution inequalities. Workable sufficient conditions are given and a comparison is made with super Poincaré inequalities.

Classification : 60E15, 26D10
Mots clés : Poincaré inequality, concentration of measure, transportation-cost inequalities, inf-convolution inequalities, logarithmic-Sobolev inequalities, super Poincaré inequalities

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