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Search: MSC category 46B06 ( Asymptotic theory of Banach spaces [See also 52A23] )

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Adamczak, Radosław; Latała, Rafał; Litvak, Alexander E.; Oleszkiewicz, Krzysztof; Pajor, Alain; Tomczak-Jaegermann, Nicole
 A Short Proof of Paouris' Inequality 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 inequalitiesCategories:46B06, 46B09, 52A23

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