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 , . denotes the usual Euclidean norm, and φ is a characteristic function on . 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 . 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.