Spherical distributions: Schoenberg (1938) revisited

An in-dimensional random vector X is said to have a spherical distribution if and only if its characteristic function is of the form phi(parallel to t parallel to), where t is an element of R-m, parallel to.parallel to denotes the usual Euclidean norm, and phi is a characteristic function on R. A more intuitive description is that the probability density function of X is constant on spheres. The class phi(m) of these characteristic functions phi is fundamental in the theory of spherical distributions on R-m. An important result, which was originally proved by Schoenberg (Ann. Math. 39(4) (1938) 811-841), is that the underlying characteristic function phi of a spherically distributed random m-vector X belongs to phi(infinity) if and only if the distribution of X is a scale mixture of normal distributions. A proof in the context of exchangeability has been given by Kingman (Biometrika 59 (1972) 492-494). Using probabilistic tools, we will give an alternative proof in the spirit of Schoenberg we think is more elegant and less complicated. (c) 2005 Elsevier GmbH. All rights reserved