A short proof of Paouris' inequality

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 $t\ge 1$, $P(|X|\ge ct\sqrt{n})\le \exp (-t\sqrt{n})$. More precisely we show that for any log-concave random vector $X$ and any $p\ge 1$, $(E|X{|}^p{)}^{1/p}\sim E|X|+{\sup }_{z\in S^{n-1}}(E|<z,X>{|}^p{)}^{1/p}$.

References in corpus (1)

• Moment estimates for convex measures

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