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Moment estimates for convex measures


 
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1. Title Title of document Moment estimates for convex measures
 
2. Creator Author's name, affiliation, country Radosław Adamczak; University of Warsaw; Poland
 
2. Creator Author's name, affiliation, country Olivier Guédon; Université Paris-Est Marne-la-Vallée; France
 
2. Creator Author's name, affiliation, country Rafał Latała; University of Warsaw; Poland
 
2. Creator Author's name, affiliation, country Alexander E. Litvak; University of Alberta; Canada
 
2. Creator Author's name, affiliation, country Krzysztof Oleszkiewicz; University of Warsaw; Poland
 
2. Creator Author's name, affiliation, country Alain Pajor; Université Paris-Est Marne-la-Vallée; France
 
2. Creator Author's name, affiliation, country Nicole Tomczak-Jaegermann; University of Alberta; Canada
 
3. Subject Discipline(s)
 
3. Subject Keyword(s) convex measures, $\kappa$-concave measure, tail inequalities, small ball probability estimate.
 
3. Subject Subject classification 46B06; 60E15; 60F10; 52A23; 52A40
 
4. Description 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 $$(\mathbb{E} |X|^{-p} )^{-1/{p}} \geq  c(\varepsilon) \left( \mathbb{E} |X| - C \sigma_{p}(X)\right) . $$
 
5. Publisher Organizing agency, location
 
6. Contributor Sponsor(s)
 
7. Date (YYYY-MM-DD) 2012-11-24
 
8. Type Status & genre Peer-reviewed Article
 
8. Type Type
 
9. Format File format PDF
 
10. Identifier Uniform Resource Identifier http://ejp.ejpecp.org/article/view/2150
 
10. Identifier Digital Object Identifier 10.1214/EJP.v17-2150
 
11. Source Journal/conference title; vol., no. (year) Electronic Journal of Probability; Vol 17
 
12. Language English=en en
 
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