  Necessary conditions for the existence of non-central Wishart distributions
are given. Our method relies on positivity properties of spherical polynomials
on Euclidean Jordan Algebras and advances an approach by Peddada and Richards
(1991), where only a special case (positive semidefinite matrices, rank one
non-centrality parameter) is treated. Not only needs the shape parameters be in
the Wallach set - as is the case for Riesz measures - but also the rank of the
non-centrality parameter is constrained by the size of the shape parameter.
This rank condition has been recently proved with different methods for the
special case of symmetric, positive semidefinite matrices (Letac and Massam
(2011) and Graczyk, Malecki and Mayerhofer (2016)).
