A Short Proof of Paouris' Inequality

作者：Adamczak Radoslaw^*; Latala Rafal; Litvak Alexander E; Oleszkiewicz Krzysztof; Pajor Alain; Tomczak Jaegermann Nicole

来源：Canadian Mathematical Bulletin, 2014, 57(1): 3-8.

DOI：10.4153/CMB-2012-014-5

• log-concave random vectors
• deviation inequalities

We give a short proof of a result of G. Paouris on the tail behaviour of the Euclidean norm vertical bar X vertical bar of an isotropic log-concave random vector X is an element of R-n, stating that for every t >= 1,
P(vertical bar X vertical bar >= ct root n) <= exp(-t root n).
More precisely we show that for any log-concave random vector X and any p >= 1,
(E vertical bar x vertical bar(P))(1/P) similar to E vertical bar X vertical bar + sup(z is an element of s)(n-1) (E vertical bar < z,X >vertical bar(p))(1/p).

• 出版日期2014-3