Schoenberg's Theorem via the law of large numbers

Davar Khoshnevisan

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Schoenberg's Theorem via the law of large numbers

Davar Khoshnevisan

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Abstract

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.

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Davar Khoshnevisan. 2005-05-02. Schoenberg's Theorem via the law of large numbers. https://arxiv.org/abs/math/0504603

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