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 $q > h(2) = 1$.