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# A Short Proof of Paouris' Inequality

Published:2012-08-25

Institute of Mathematics, University of Warsaw, Banacha 2, 02-097 Warszawa, Poland
• Rafał Latała,
Institute of Mathematics, University of Warsaw, Banacha 2, 02-097 Warszawa, Poland
• Alexander E. Litvak,
Dept. of Math. and Stat. Sciences, University of Alberta, Edmonton, AB T6G 2G1
• Krzysztof Oleszkiewicz,
Institute of Mathematics, University of Warsaw, Banacha 2, 02-097 Warszawa, Poland
• Alain Pajor,
Université Paris-Est, Équipe d'Analyse et Mathématiques Appliquées, 5, boulevard Descartes, Champs sur Marne, 77454 Marne-la-Vallée, Cedex 2, France
• Nicole Tomczak-Jaegermann,
Dept. of Math. and Stat. Sciences, University of Alberta, Edmonton, AB T6G 2G1
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## Abstract

We give a short proof of a result of G.~Paouris on the tail behaviour of the Euclidean norm $|X|$ of an isotropic log-concave random vector $X\in\mathbb{R}^n,$ stating that for every $t\geq 1$, $\mathbb{P} \big( |X|\geq ct\sqrt n\big)\leq \exp(-t\sqrt n).$ More precisely we show that for any log-concave random vector $X$ and any $p\geq 1$, $(\mathbb{E}|X|^p)^{1/p}\sim \mathbb{E} |X|+\sup_{z\in S^{n-1}}(\mathbb{E} |\langle z,X\rangle|^p)^{1/p}.$
 Keywords: log-concave random vectors, deviation inequalities