Geodesic
A Short Proof of Paouris' Inequality
Canadian mathematical bulletin, Tome 57 (2014) no. 1, pp. 3-8

Voir la notice de l'article provenant de la source Cambridge

DOI

We give a short proof of a result of G. Paouris on the tail behaviour of the Euclidean norm $\left| X \right|$ of an isotropic log-concave random vector $X\,\in \,{{\mathbb{R}}^{n}},$ stating that for every $t\,\ge \,1$ , $$\mathbb{P}\left( \left| X \right|\,\ge \,ct\sqrt{n} \right)\,\le \,\exp (-t\sqrt{n}).$$ More precisely we show that for any log-concave random vector $X$ and any $p\,\ge \,1$ , $${{(\mathbb{E}{{\left| X \right|}^{p}})}^{1/p}}\,\sim \,\mathbb{E}\left| X \right|\,+\,\underset{z\in {{S}^{n-1}}}{\mathop{\sup }}\,\,{{(\mathbb{E}{{\left| \left\langle z,\,X \right\rangle\right|}^{p}})}^{1/p}}.$$
DOI : 10.4153/CMB-2012-014-5
Mots-clés : 46B06, 46B09, 52A23, log-concave random vectors, deviation inequalities 
Adamczak, Radosław; Latała, Rafał; Litvak, Alexander E.; Oleszkiewicz, Krzysztof; Pajor, Alain; Tomczak-Jaegermann, Nicole. A Short Proof of Paouris' Inequality. Canadian mathematical bulletin, Tome 57 (2014) no. 1, pp. 3-8. doi: 10.4153/CMB-2012-014-5
@article{10_4153_CMB_2012_014_5,
     author = {Adamczak, Rados{\l}aw and Lata{\l}a, Rafa{\l} and Litvak, Alexander E. and Oleszkiewicz, Krzysztof and Pajor, Alain and Tomczak-Jaegermann, Nicole},
     title = {A {Short} {Proof} of {Paouris'} {Inequality}},
     journal = {Canadian mathematical bulletin},
     pages = {3--8},
     year = {2014},
     volume = {57},
     number = {1},
     doi = {10.4153/CMB-2012-014-5},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-2012-014-5/}
}
TY  - JOUR
AU  - Adamczak, Radosław
AU  - Latała, Rafał
AU  - Litvak, Alexander E.
AU  - Oleszkiewicz, Krzysztof
AU  - Pajor, Alain
AU  - Tomczak-Jaegermann, Nicole
TI  - A Short Proof of Paouris' Inequality
JO  - Canadian mathematical bulletin
PY  - 2014
SP  - 3
EP  - 8
VL  - 57
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CMB-2012-014-5/
DO  - 10.4153/CMB-2012-014-5
ID  - 10_4153_CMB_2012_014_5
ER  - 
%0 Journal Article
%A Adamczak, Radosław
%A Latała, Rafał
%A Litvak, Alexander E.
%A Oleszkiewicz, Krzysztof
%A Pajor, Alain
%A Tomczak-Jaegermann, Nicole
%T A Short Proof of Paouris' Inequality
%J Canadian mathematical bulletin
%D 2014
%P 3-8
%V 57
%N 1
%U http://geodesic.mathdoc.fr/articles/10.4153/CMB-2012-014-5/
%R 10.4153/CMB-2012-014-5
%F 10_4153_CMB_2012_014_5

Cité par Sources :