On the parameter domain of Wishart distributions and their infinite divisibility

TL;DR

This paper fully characterizes Wishart distributions on positive semi-definite matrix cones, including degenerate cases, and links infinitely divisible Wishart distributions to gamma distributions, extending classical results.

Contribution

It provides a new complete description of the parameter domain for Wishart distributions, including degenerate scale parameters, and connects infinite divisibility to gamma distributions.

Findings

Complete characterization of Wishart parameter domain.

Inclusion of degenerate scale parameters in the analysis.

Wishart distributions' infinite divisibility linked to gamma distributions.

Abstract

A complete characterization of Wishart distributions on the cones of positive semi-definite matrices is provided in terms of a description of their maximal parameter domain. This result is new in that also degenerate scale parameters are included. For such cases, the standard constraints on the range of the shape parameter may be relaxed. Furthermore, the infinitely divisible Wishart distributions are revealed as suitable transformations and embeddings of one dimensional gamma distributions. This note completes the findings of L\'evy (1937) concerning infinite divisibility and Gindikin (1975) regarding the existence issue.