Moment estimates for convex measures

Abstract

Let $p\ge 1$, $\eps >0$, $r\geq (1+\eps) p$, and $X$ be a $(-1/r)$-concave random vector in ${\mathbb{R}}^n$ with Euclidean norm $|X|$. We prove that $(\E |X|^{p})^{1/{p}}\leq c (C(\eps) \E|X|+σ_{p}(X))$, where $σ_{p}(X)=\sup_{|z|\leq 1}(\E|<z,X>|^{p})^{1/p}$, $C(\eps)$ depends only on $\eps$ and $c$ is a universal constant. Moreover, if in addition $X$ is centered then $(\E |X|^{-p})^{-1/{p}}\geq c(\eps) (\E|X| - C σ_{p}(X))$.

References in corpus (1)

• A short proof of Paouris' inequality

Cited by in corpus (4)

• A short proof of Paouris' inequality
• Concentration of information content for convex measures
• Small ball estimates for quasi-norms
• Estimating the covariance of random matrices