A short proof of Paouris' inequality

TL;DR

This paper provides a concise proof of Paouris' inequality, which describes the tail decay of the Euclidean norm of isotropic log-concave vectors, highlighting their concentration properties.

Contribution

It offers a simplified proof of Paouris' inequality and extends understanding of the moments and tail behavior of log-concave random vectors.

Findings

Tail probability of Euclidean norm decays exponentially with dimension

Moment equivalence for log-concave vectors and their projections

Simplified proof technique for 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 $X\in\R^n$, stating that for every $t\geq 1$, $P(|X|\geq ct\sqrt n)\leq \exp(-t\sqrt n)$. More precisely we show that for any log-concave random vector $X$ and any $p\geq 1$, $(E|X|^p)^{1/p}\sim E |X|+\sup_{z\in S^{n-1}}(E |< z,X>|^p)^{1/p}$.