Convex and star-shaped sets associated with stable distributions

Abstract

It is known that each symmetric stable distribution in $R^d$ is related to a norm on $R^d$ that makes $R^d$ embeddable in $L_p([0,1])$. In case of a multivariate Cauchy distribution the unit ball in this norm corresponds is the polar set to a convex set in $R^d$ called a zonoid. This work interprets general 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 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. It is shown that each symmetric stable laws appears as the limit for the sum of sub-Gaussian laws and an estimate for the probability distance to a sub-Gaussian law is given. Operations with convex sets induce the well-known and new operations with stable vectors. Furthermore, covariation, regression and orthogonality concepts for stable laws acquire geometric interpretations. A similar collection of results is presented for one-sided stable laws.