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 $\overline{{\mathcal{P}}_d}$ 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 $\overline{{\mathcal{P}}_d}$ and with Laplace transform $(\det s)^{-n/2}\exp \tr(s^{-1}w)$where$n$isanintegerandwhere$w=$\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}$(0,…,0,1,…,1)$hasorder$d$andrank$k.$Wecompute$m(d-1,d,d)$andweshowthatneither$m(d-2,d,d)$nor$m(d-2,d-1,d)$ exist. This proves a conjecture of E. Mayerhofer.