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A Short Proof of Paouris' Inequality

  • Radosław Adamczak,
    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 log-concave random vectors, deviation inequalities
MSC Classifications: 46B06, 46B09, 52A23 show english descriptions Asymptotic theory of Banach spaces [See also 52A23]
Probabilistic methods in Banach space theory [See also 60Bxx]
Asymptotic theory of convex bodies [See also 46B06]
46B06 - Asymptotic theory of Banach spaces [See also 52A23]
46B09 - Probabilistic methods in Banach space theory [See also 60Bxx]
52A23 - Asymptotic theory of convex bodies [See also 46B06]
 

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