It is shown that every symmetric $\alpha$-stable random variable $X, 0 < \alpha \leq 2$, has the property: For any $p$ and $q, 0 \leq h(\alpha) < q < p < \alpha$, there is a constant $s > 0$ such that $(E\|x + sXy\|^p)^{1/p} \leq (E\|x + Xy\|^q)^{1/q},$ for all $x$ and $y$ from a normed space. The quantity $h(\alpha)$ is identically 0 if $\alpha \leq 1$. It is strictly less than 1 for every $\alpha < 2$ which reveals the contrast to the Gaussian case in which $q > h(2) = 1$.
Publié le : 1990-10-14
Classification:
Hypercontraction,
stable random variables,
domain of normal attraction,
normed space,
60E07,
60B11,
60E15,
43A15,
42C05
@article{1176990645,
author = {Szulga, Jerzy},
title = {A Note on Hypercontractivity of Stable Random Variables},
journal = {Ann. Probab.},
volume = {18},
number = {4},
year = {1990},
pages = { 1746-1758},
language = {en},
url = {http://dml.mathdoc.fr/item/1176990645}
}
Szulga, Jerzy. A Note on Hypercontractivity of Stable Random Variables. Ann. Probab., Tome 18 (1990) no. 4, pp. 1746-1758. http://gdmltest.u-ga.fr/item/1176990645/