Infimum-convolution description of concentration properties of product probability measures, with applications

This paper is devoted to the concentration properties of product probability measures mu = mu(1) circle times(...)circle times mu(n), expressed in term of dimension-free functional inequalities of the form [GRAPHICS] where a is a parameter, 0 < alpha < 1, and Q(alpha)f is an appropriate infimum-convolution operator. This point of view has been introduced by Maurey [B. Maurey, Some deviation inequalities, Geom. Funct: Anal. 1 (1991) 188-197]. It has its origins in concentration inequalities by Talagrand where the enlargement of sets is done in accordance with the cost function of the operator Q alpha f (see [M. Talagrand, Concentration of measure and isoperimetric, inequalities in product spaces, Publ. Math. Inst. Hautes Etudes Sci. 81 (1995) 73-205, M. Talagrand, New concentration inequalities in product spaces, Invent. Math. 126 *(1996) 505-563, M. Talagrand, A new look at independence, Ann. Probab. 24 (1996) 1-34]). A main application of the functional inequalities obtained here is optimal deviations inequalities for suprema of sums of independent random variables. As example, we also derive classical deviations bounds for the one-dimensional bin packing problem. (c) 2006 Elsevier Masson SAS. All rights reserved.