Convex and star-shaped sets associated with multivariate stable distributions, I: Moments and densities

It is known that each symmetric stable distribution in RdRd is related to a norm on RdRd that makes RdRd embeddable in Lp([0,1])Lp([0,1]). In the case of a multivariate Cauchy distribution the unit ball in this norm is the polar set to a convex set in RdRd called a zonoid. This work interprets symmetric stable laws using convex or star-shaped sets and exploits recent advances in convex geometry in order to come up with new probabilistic results for multivariate symmetric stable distributions. In particular, it provides expressions for moments of the Euclidean norm of a stable vector, mixed moments and various integrals of the density function. It is shown how to use geometric inequalities in order to bound important parameters of stable laws. Furthermore, covariation, regression and orthogonality concepts for stable laws acquire geometric interpretations.