A characterization of dimension free concentration in terms of transportation inequalities

Details A characterization of dimension free concentration in terms of transportation inequalities A characterization of dimension free concentration in terms of transportation inequalities

2008-04-18

arxiv.org/abs/0804.3089v1

Ann. Probab. 37 (2009), no. 6, 2480-2498

Mathematics

Probability

18 p

Scientific paper

10.1214/09-AOP470

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 Poincar\'e inequality is equivalent to a certain form of dimension free exponential concentration. The proofs of these results rely on simple Large Deviations techniques.

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