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A Short Proof of Paouris' Inequality


Published:20120825
Printed: Mar 2014
Radosław Adamczak,
Institute of Mathematics, University of Warsaw, Banacha 2, 02097 Warszawa, Poland
Rafał Latała,
Institute of Mathematics, University of Warsaw, Banacha 2, 02097 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, 02097 Warszawa, Poland
Alain Pajor,
Université ParisEst, Équipe d'Analyse et Mathématiques Appliquées, 5, boulevard Descartes, Champs sur Marne, 77454 MarnelaVallée, Cedex 2, France
Nicole TomczakJaegermann,
Dept. of Math. and Stat. Sciences, University of Alberta, Edmonton, AB T6G 2G1
Abstract
We give a short proof of a result of G.~Paouris on
the tail behaviour of the Euclidean norm $X$ of an isotropic
logconcave 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 logconcave random vector $X$
and any $p\geq 1$,
\[(\mathbb{E}X^p)^{1/p}\sim \mathbb{E} X+\sup_{z\in
S^{n1}}(\mathbb{E} \langle
z,X\rangle^p)^{1/p}.\]