Moment estimates for convex measures

TL;DR

This paper establishes bounds on the moments of convex measures, specifically for $(-1/r)$-concave random vectors, providing inequalities that relate the moments of the Euclidean norm to other statistical measures.

Contribution

It introduces new moment inequalities for convex measures, extending understanding of their probabilistic behavior and providing bounds that depend only on specific parameters.

Findings

Derived upper bounds for positive moments of convex measures.

Established lower bounds for negative moments of convex measures.

Results depend only on parameters $\eps$, $p$, and universal constants.

Abstract

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