A Short Proof of Paouris' Inequality

Adamczak, Radosław ; Latała, Rafał ; Litvak, Alexander E. ; Oleszkiewicz, Krzysztof ; Pajor, Alain ; Tomczak-Jaegermann, Nicole

Canadian mathematical bulletin, Tome 57 (2014) no. 1, pp. 3-8

Résumé

We give a short proof of a result of G. Paouris on the tail behaviour of the Euclidean norm $\left| X \right|$ of an isotropic log-concave random vector $X\,\in \,{{\mathbb{R}}^{n}},$ stating that for every $t\,\ge \,1$ , $$\mathbb{P}\left( \left| X \right|\,\ge \,ct\sqrt{n} \right)\,\le \,\exp (-t\sqrt{n}).$$ More precisely we show that for any log-concave random vector $X$ and any $p\,\ge \,1$ , $${{(\mathbb{E}{{\left| X \right|}^{p}})}^{1/p}}\,\sim \,\mathbb{E}\left| X \right|\,+\,\underset{z\in {{S}^{n-1}}}{\mathop{\sup }}\,\,{{(\mathbb{E}{{\left| \left\langle z,\,X \right\rangle\right|}^{p}})}^{1/p}}.$$

Détail

DOI : 10.4153/CMB-2012-014-5

Mots-clés : 46B06, 46B09, 52A23, log-concave random vectors, deviation inequalities

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