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Concentration of measure and logarithmic Sobolev inequalities

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Concentration of measure and logarithmic Sobolev inequalities SÉMINAIRE DE PROBABILITÉS (STRASBOURG) MICHEL LEDOUX Concentration of measure and logarithmic Sobolev inequalities Séminaire de probabilités (Strasbourg), tome 33 (1999), p. 120-216. <http://www.numdam.org/item?id=SPS_1999__33__120_0> © Springer-Verlag, Berlin Heidelberg New York, 1999, tous droits réservés. L’accès aux archives du séminaire de probabilités (Strasbourg) (http://www-irma. u-strasbg.fr/irma/semproba/index.shtml), implique l’accord avec les conditions gé- nérales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commer- ciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ CONCENTRATION OF MEASURE AND LOGARITHMIC SOBOLEV INEQUALITIES MICHEL LEDOUX TABLE OF CONTENTS INTRODUCTION 123 1. ISOPERIMETRIC AND CONCENTRATION INEQUALITIES 126 1.1 Introduction 126 1.2 Isoperimetric inequalities for Gaussian and Boltzmann measures 127 1.3 Some general facts about concentration 134 2. SPECTRAL GAP AND LOGARITHMIC SOBOLEV INEQUALITIES 139 2.1 Abstract functional inequalities 139 2.2 Examples of logarithmic Sobolev inequalities 145 2.3 Herbst’s argument 148 2.4 Entropy-energy inequalities and non-Gaussian tails 154 2.5 Poincare inequalities and concentration 159 3. DEVIATION INEQUALITIES FOR PRODUCT MEASURES 161 3.1 Concentration with respect to the Hamming metric 161 3.2 Deviation inequalities for convex functions 163 3.3 Information inequalities and concentration 166 3.4 Applications to bounds on empirical processes 171 4. MODIFIED LOGARITHMIC SOBOLEV INEQUALITIES FOR LOCAL GRADIENTS 173 4.1 The exponential measure 173 4.2 Modified logarithmic Sobolev inequalities 178 4.3 Poincare inequalities and modified logarithmic Sobolev inequalities 179 5. MODIFIED LOGARIT

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