Absolute Continuity of Stable Seminorms

Abstract

Suppose that $E$ is a complete separable real metric vector space. It is proved that if $X$ is a symmetric $E$-valued $p$-stable random vector, $0 < p < 2$, and $q$ is a lower semicontinuous, a.s. finite seminorm, then the distribution of $q(X)$ is absolutely continuous apart from a possible jump. If, additionally, $q$ is strictly convex or $0 < p < 1$, then the distribution of $q(X)$ is either absolutely continuous or degenerate at 0. This result settles, in particular, the problem of absolute continuity of the supremum of stable sequences, extending thus Tsirel'son's theorem.