## The Annals of Probability

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

### A Note on Hypercontractivity of Stable Random Variables

#### 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

**Permanent link to this document**

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**

links.jstor.org

**Subjects**

Primary: 60E07: Infinitely divisible distributions; stable distributions

Secondary: 60B11: Probability theory on linear topological spaces [See also 28C20] 60E15: Inequalities; stochastic orderings 43A15: $L^p$-spaces and other function spaces on groups, semigroups, etc. 42C05: Orthogonal functions and polynomials, general theory [See also 33C45, 33C50, 33D45]

**Keywords**

Hypercontraction stable random variables domain of normal attraction normed space

#### Citation

Szulga, Jerzy. A Note on Hypercontractivity of Stable Random Variables. Ann. Probab. 18 (1990), no. 4, 1746--1758. doi:10.1214/aop/1176990645. https://projecteuclid.org/euclid.aop/1176990645