| Título: | Moment estimates for convex measures |
| Autores: |
Adamczak, Radosław; University of Warsaw Guédon, Olivier; Université Paris-Est Marne-la-Vallée Latała, Rafał; University of Warsaw Litvak, Alexander E.; University of Alberta Oleszkiewicz, Krzysztof; University of Warsaw Pajor, Alain; Université Paris-Est Marne-la-Vallée Tomczak-Jaegermann, Nicole; University of Alberta |
| Fecha: | 2012-01-01 |
| Publicador: | Electronic journal of probability |
| Fuente: |
 |
| Tipo: | Peer-reviewed Article |
| Tema: |
convex measures, $\kappa$-concave measure, tail inequalities, small ball probability estimate. 46B06; 60E15; 60F10; 52A23; 52A40 |
| Descripción: | 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 $$(\mathbb{E} |X|^{-p} )^{-1/{p}} \geq c(\varepsilon) \left( \mathbb{E} |X| - C \sigma_{p}(X)\right) . $$ |
| Idioma: | Inglés |
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