On Wishart and non-central Wishart distributions on symmetric cones

TL;DR

This paper establishes necessary conditions for the existence of non-central Wishart distributions on symmetric cones, extending previous results by using positivity properties of spherical polynomials in Euclidean Jordan Algebras.

Contribution

It generalizes existing conditions for Wishart distributions beyond positive semidefinite matrices to symmetric cones using Jordan algebra techniques.

Findings

Shape parameters must be in the Wallach set

Rank of non-centrality parameter is constrained by shape parameter

Extends previous results to general symmetric cones

Abstract

Necessary conditions for the existence of non-central Wishart distributions are given. Our method relies on positivity properties of spherical polynomials on Euclidean Jordan Algebras and advances an approach by Peddada and Richards (1991), where only a special case (positive semidefinite matrices, rank one non-centrality parameter) is treated. Not only needs the shape parameters be in the Wallach set - as is the case for Riesz measures - but also the rank of the non-centrality parameter is constrained by the size of the shape parameter. This rank condition has been recently proved with different methods for the special case of symmetric, positive semidefinite matrices (Letac and Massam (2011) and Graczyk, Malecki and Mayerhofer (2016)).