# Non-central Wishart as mixture of central Wisharts?

The non-central chi-square distribution with $\nu$ degrees of freedom has a density which can be expressed as $$f(x) = \sum_{i=0}^{\infty} c_i f_{\nu + 2i}(x),$$ where $c_i$ is a function of the non-centrality parameter and $i$, and where $f_{j}(x)$ is the density of a chi-square distribution with $j$ degrees of freedom. (The non-central chi-squared distribution is a Poisson weighted mixture of central chi-squared distributions).

My question is whether this property extends to the multivariate case, the non-central Wishart distribution: can we express the non-central Wishart as a weighted mixture of central Wishart distributions?

(My guess is not, except in the scalar case; if it does hold, however, and the $c_i$ are easy to compute, I would like to use this property to compute the moments of the non-central Wishart.)

• It looks like Theorem 1 of Li and Geng provides something like what I would need, for the case where the non-centrality matrix is rank 1. Mar 23 '18 at 4:53