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arXiv · 0804.3089

A characterization of dimension free concentration in terms of transportation inequalities

Nathael Gozlan
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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 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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Nathael Gozlan. 2008-04-18. A characterization of dimension free concentration in terms of transportation inequalities. https://doi.org/10.1214/09-aop470

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