## The Annals of Probability

### A Note on Hypercontractivity of Stable Random Variables

Jerzy Szulga

#### 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 $q > h(2) = 1$.

#### Article information

Source
Ann. Probab., Volume 18, Number 4 (1990), 1746-1758.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176990645

Digital Object Identifier
doi:10.1214/aop/1176990645

Mathematical Reviews number (MathSciNet)
MR1071822

Zentralblatt MATH identifier
0716.60016

JSTOR