Concentration of measure principle and entropy-inequalities

Paul-Marie Samson (1)

1 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées

The concentration measure principle is presented in an abstract way to encompass and unify different concentration properties. We give a general overview of the links between concentration properties, transport-entropy inequalities, and logarithmic Sobolev inequalities for some specific transport costs. By giving few examples, we emphasize optimal weak transport costs as an efficient tool to establish new transport inequality and new concentration principles for discrete measures (the binomial law, the Poisson measure, the uniform law on the symmetric group).

Mots clés: Logarithmic-Sobolev inequalities; Duality; Transport inequalities; Concentration of measure

Domaines: Probabilités [math.PR]; Analyse fonctionnelle [math.FA]

Paul-Marie Samson. Concentration of measure principle and entropy-inequalities . Eric Carlen, Mokshay Madiman, Elisabeth M. Werner. Convexity and Concentration, 161 (Chapter 3), Springer, New York, NY, 2017, The IMA Volumes in Mathematics and its Applications, 9781493970049 • 9781493970056 (online). ⟨10.1007/978-1-4939-7005-6⟩. ⟨hal-01365314⟩