An extension of Wick's theorem

TL;DR

This paper extends Wick's theorem to elliptical distributions, including Gaussian scale mixtures, enabling easier polynomial integration over spheres and broadening its applicability in statistical theory.

Contribution

The paper introduces a novel extension of Wick's theorem to elliptical distributions, expanding its theoretical scope and practical applications.

Findings

Wick's theorem extended to uniform sphere distribution

Extension to all elliptical distributions including Gaussian scale mixtures

Simplified proof of Folland's theorem on polynomial integration over spheres

Abstract

We propose an extension of a result by Repetowicz et al. about Wick's theorem and its applications: we first show that Wick's theorem can be extended to the uniform distribution on the sphere and then to the whole class of elliptical distributions. Then, as a special case, we detail this result for distributions that are Gaussian scale mixtures. Finally, we show that these results allow to recover easily a theorem by Folland about integration of polynomials over the sphere.