Preprint:
Schoenberg's Theorem via the Law of Large Numbers
(NOT FOR PUBLICATION)
D. Khoshnevisan
Abstract.
A classical theorem of S. Bochner states that
a function f : R→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 converse to Bochner's theorem. We present
a nontechnical 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)
60Fxx, 43A35
Support. Research supported in part by a grant from
The National Science Foundation.
Pre/EPrints. This paper is available in
Davar Khoshnevisan
Department of Mathematics
University of Utah
155 S, 1400 E JWB 233
Salt Lake City, UT 841120090, U.S.A.
davar@math.utah.edu

Last Update: May 2, 2005
© 2005  Davar Khoshnevisan