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2012 Moment estimates for convex measures
Radosław Adamczak, Olivier Guédon, Rafał Latała, Alexander Litvak, Krzysztof Oleszkiewicz, Alain Pajor, Nicole Tomczak-Jaegermann
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Electron. J. Probab. 17: 1-19 (2012). DOI: 10.1214/EJP.v17-2150

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 $$(\mathbb{E} |X|^{-p} )^{-1/{p}} \geq c(\varepsilon) \left( \mathbb{E} |X| - C \sigma_{p}(X)\right) . $$

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Radosław Adamczak. Olivier Guédon. Rafał Latała. Alexander Litvak. Krzysztof Oleszkiewicz. Alain Pajor. Nicole Tomczak-Jaegermann. "Moment estimates for convex measures." Electron. J. Probab. 17 1 - 19, 2012. https://doi.org/10.1214/EJP.v17-2150

Information

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

zbMATH: 1286.46014
MathSciNet: MR3005719
Digital Object Identifier: 10.1214/EJP.v17-2150

Subjects:
Primary: 46B06
Secondary: 52A23, 52A40, 60E15, 60F10

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