Spherical distributions: Schoenberg (1938) revisited

An m-dimensional random vector X is said to have a spherical distribution if and only if its characteristic function is of the form φ(∥t∥), where t∈Rm, ∥.∥ denotes the usual Euclidean norm, and φ is a characteristic function on R. A more intuitive description is that the probability density function of X is constant on spheres. The class Φm of these characteristic functions φ is fundamental in the theory of spherical distributions on Rm. An important result, which was originally proved by Schoenberg (Ann. Math. 39(4) (1938) 811-841), is that the underlying characteristic function φ of a spherically distributed random m-vector X belongs to Φ∞ 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.