Existence and non-existence of the non-central Wishart distributions

The problem considered in this paper is to find when the non-central Wishart distribution, defined on the cone $\bar{\mathcal{P}_d}$ of semi positive definite matrices of order $d$ and with a real valued shape parameter, exists. We reduce this problem to the problem of existence of the measures $m(n,k,d)$ defined on $\bar{\mathcal{P}_d}$ and with Laplace transform $(\det s)^{-n/2}\exp \tr(s^{-1}w)$ where $n$ is an integer and where $w=\mathrm{diag}(0,...,0,1,...,1)$ has order $d$ and rank $k.$ We compute $m(d-1,d,d)$ and we show that neither $m(d-2,d,d)$ nor $m(d-2,d-1,d)$ exist. This proves a conjecture of E. Mayerhofer.