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October, 1990 A Note on Hypercontractivity of Stable Random Variables
Jerzy Szulga
Ann. Probab. 18(4): 1746-1758 (October, 1990). DOI: 10.1214/aop/1176990645

Abstract

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 $q > h(2) = 1$.

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Jerzy Szulga. "A Note on Hypercontractivity of Stable Random Variables." Ann. Probab. 18 (4) 1746 - 1758, October, 1990. https://doi.org/10.1214/aop/1176990645

Information

Published: October, 1990
First available in Project Euclid: 19 April 2007

zbMATH: 0716.60016
MathSciNet: MR1071822
Digital Object Identifier: 10.1214/aop/1176990645

Subjects:
Primary: 60E07
Secondary: 42C05 , 43A15 , 60B11 , 60E15

Keywords: domain of normal attraction , Hypercontraction , normed space , Stable random variables

Rights: Copyright © 1990 Institute of Mathematical Statistics

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Vol.18 • No. 4 • October, 1990
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