The Laplace transform (dets)−pexptr(s−1w) and the existence of non-central Wishart distributions

The problem considered in this paper is to find when the non-central Wishart distribution, defined on the cone Pd¯ of positive semidefinite matrices of order d and with a real-valued shape parameter p, does exist. This can be reduced to the study of the measures m(n,k,d) defined on Pd¯ and with Laplace transform (dets)−n∕2exptr(s−1w), where n is an integer and w=diag(0,…,0,1,…,1) has order d and rank k. Our two main results are the following: we compute m(d−1,d,d) and we show that neither m(d−2,d,d) nor m(d−2,d−1,d) exists. These facts solve the problems of the existence and computation of these non-central Wishart distributions.