Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Poincaré inequalities and dimension free concentration of measure

Nathael Gozlan

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Abstract

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.

Résumé

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.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 46, Number 3 (2010), 708-739.

Dates
First available in Project Euclid: 6 August 2010

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1281100396

Digital Object Identifier
doi:10.1214/09-AIHP209

Mathematical Reviews number (MathSciNet)
MR2682264

Zentralblatt MATH identifier
1205.60040

Subjects
Primary: 60E15: Inequalities; stochastic orderings 26D10: Inequalities involving derivatives and differential and integral operators

Keywords
Poincaré inequality Concentration of measure Transportation-cost inequalities Inf-convolution inequalities Logarithmic-Sobolev inequalities Super Poincaré inequalities

Citation

Gozlan, Nathael. Poincaré inequalities and dimension free concentration of measure. Ann. Inst. H. Poincaré Probab. Statist. 46 (2010), no. 3, 708--739. doi:10.1214/09-AIHP209. https://projecteuclid.org/euclid.aihp/1281100396


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