Schoenberg's Theorem via the Law of Large Numbers

D. Khoshnevisan

Abstract. A classical theorem of S. Bochner states that a function f : RC is the Fourier transform of a finite Borel measure if and only if f is positive definite. In 1938, I. Schoenberg found a beautiful converse to Bochner's theorem. We present a non-technical derivation of Schoenberg's theorem that relies chiefly on the de Finetti theorem and the law of large numbers of classical probability theory.

Keywords. Schoenberg's theorem, Law of large numbers

AMS Classification (2000) 60F-xx, 43A35

Support. Research supported in part by a grant from The National Science Foundation.

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Davar Khoshnevisan
Department of Mathematics
University of Utah
155 S, 1400 E JWB 233
Salt Lake City, UT 84112-0090, U.S.A.

Last Update: May 2, 2005
© 2005 - Davar Khoshnevisan