Electronic Journal of Probability

Moment estimates for convex measures

Radosław Adamczak, Olivier Guédon, Rafał Latała, Alexander Litvak, Krzysztof Oleszkiewicz, Alain Pajor, and Nicole Tomczak-Jaegermann

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 46B06: Asymptotic theory of Banach spaces [See also 52A23]
Secondary: 60E15: Inequalities; stochastic orderings 60F10: Large deviations 52A23: Asymptotic theory of convex bodies [See also 46B06] 52A40: Inequalities and extremum problems

convex measures $\kappa$-concave measure tail inequalities small ball probability estimate

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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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