Compact group actions, spherical bessel functions, and invariant random variables

The theory of compact group actions on locally compact abelian groups provides a unifying theory under which different invariance conditions studied in several contexts by a number of statisticians are subsumed as special cases. For example, Schoenberg's characterization of radially symmetric characteristic functions on n is extended to this general context and the integral representations are expressed in terms of the generalized spherical Bessel functions of Gross and Kunze. These same Bessel functions are also used to obtain a variant of the Lévy-Khinchine formula of Parthasarathy, Ranga Rao, and Varadhan appropriate to invariant distributions.