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2 Bulletin of the Polish Academy of Sciences. Mathematics

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On measure concentration of vector-valued maps

Ledoux M., Oleszkiewicz K.

Bulletin of the Polish Academy of Sciences. Mathematics

2007

Vol. 55, no 3

261-278

We study concentration properties for vector-valued maps. In particular, we describe inequalities which capture the exact dimensional behavior of Lipschitz maps with values in R^k. To this end, we study in particular a domination principle for projections which might be of independent interest. We further compare our conclusions with earlier results by Pinelis in the Gaussian case, and discuss extensions to the infinite-dimensional setting.

Logarithmic Sobolev inequalities and concentration of measure for convex functions and polynomial chaoses

Adamczak R.

Bulletin of the Polish Academy of Sciences. Mathematics

2005

Vol. 53, no 2

221-238

We prove logarithmic Sobolev inequalities and concentration results for convex functions and a class of product random vectors. The results are used to derive tail and moment inequalities for chaos variables (in the spirit of Talagrand and Arcones-Gine). We also show that the same proof may be used for chaoses generated by log-concave random variables, recovering results by Łochowski, and present an application to exponential integrability of Rademacher chaos.