A characterization of dimension free concentration in terms of transportation inequalities

Abstract

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.

18 p

References in corpus (3)

• Transportation cost-information inequalities and applications to random dynamical systems and diffusions
• Between Sobolev and Poincaré
• On the infimum convolution inequality

Cited by in corpus (17)

• Concentration for Coulomb gases and Coulomb transport inequalities
• Some results on random circulant matrices
• Characterization of Talagrand's transport-entropy inequalities in metric spaces
• A new characterization of Talagrand's transport-entropy inequalities and applications
• On fine properties of mixtures with respect to concentration of measure and Sobolev type inequalities
• Rates of convergence for empirical spectral measures: a soft approach
• Validated Variational Inference via Practical Posterior Error Bounds
• Functional inequalities for forward and backward diffusions
• Note on the $q$-Logarithmic Sobolev and $p$-Talagrand Inequalities on Carnot Groups
• Hydrodynamic limit for conservative spin systems with super-quadratic, partially inhomogeneous single-site potential
• Transportation cost-information and concentration inequalities for bifurcating Markov chains
• Concentration and convergence rates for spectral measures of random matrices
• Transportation-cost inequalities for diffusions driven by Gaussian processes
• Properties of Isoperimetric, Functional and Transport-Entropy Inequalities Via Concentration
• Almost sure convergence in quantum spin glasses
• Transport inequalities for random point measures
• Functional inequalities for two-level concentration