Schoenberg's Theorem via the law of large numbers
Abstract
Description
A classical theorem of S. Bochner states that a function
$f:R^n \to C$ is the Fourier transform of a finite Borel measure if and only if $f$ is positive definite. In 1938, I. Schoenberg found a beautiful complement to Bochner's theorem.
We present a non-technical derivation of of Schoenberg's theorem that relies chiefly on the de Finneti theorem and the law of large numbers of classical probability theory.
Some errors and misprints corrected; new references have been added
Some errors and misprints corrected; new references have been added