Schoenberg's Theorem via the law of large numbers
  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. 
Davar Khoshnevisan