Probability, Statistics and Planet Earth, II:The Bochner-Godement theorem for symmetric spaces

The Bochner-Godement theorem extends the classical Bochner and Bochner-Schoenberg theorems from the context of Euclidean spaces and spheres to general symmetric spaces. We show how it also includes recent results on products of symmetric spaces: the Berg-Porcu theorem (sphere cross line), and the Guella-Menegatto-Peron theorem (products of spheres), and discuss related results and applications.