Electronic Journal of Probability

Moment estimates for convex measures

Abstract

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

Article information

Source
Electron. J. Probab., Volume 17 (2012), paper no. 101, 19 pp.

Dates
Accepted: 24 November 2012
First available in Project Euclid: 4 June 2016

https://projecteuclid.org/euclid.ejp/1465062423

Digital Object Identifier
doi:10.1214/EJP.v17-2150

Mathematical Reviews number (MathSciNet)
MR3005719

Zentralblatt MATH identifier
1286.46014

Rights

Citation

Adamczak, Radosław; Guédon, Olivier; Latała, Rafał; Litvak, Alexander; Oleszkiewicz, Krzysztof; Pajor, Alain; Tomczak-Jaegermann, Nicole. Moment estimates for convex measures. Electron. J. Probab. 17 (2012), paper no. 101, 19 pp. doi:10.1214/EJP.v17-2150. https://projecteuclid.org/euclid.ejp/1465062423

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