On the existence of non-central Wishart distributions

TL;DR

This paper investigates the existence conditions of non-central Wishart distributions using affine Markov processes, confirming Eaton's conjecture on the shape parameter but showing additional constraints are needed beyond the Gindikin ensemble.

Contribution

It introduces a new method based on affine Markov processes to derive necessary conditions for non-central Wishart distributions, extending previous theoretical understanding.

Findings

Confirmed Eaton's conjecture on the shape parameter range.

Discovered that the shape parameter must satisfy more conditions than belonging to the Gindikin ensemble.

Provided joint necessary conditions on shape and non-centrality parameters.

Abstract

This paper deals with the existence issue of non-central Wishart distributions which is a research topic initiated by Wishart (1928), and with important contributions by e.g., L\'evy (1937), Gindikin (1975), Shanbhag (1988), Peddada and Richards (1991). We present a new method involving the theory of affine Markov processes, which reveals joint necessary conditions on shape and non-centrality parameter. While Eaton's conjecture concerning the necessary range of the shape parameter is confirmed, we also observe that it is not sufficient anymore that it only belongs to the Gindikin ensemble, as is in the central case.