On wishart and noncentral wishart distributions on symmetric cones

Necessary conditions for the existence of noncentral 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 [Ann. Probab. 19 (1991), pp. 868–874] where only a special case (positive semidefinite matrices, rank 1 noncentrality parameter) is treated. Not only do the shape parameters need to be in the Wallach set—as is the case for Riesz measures—but also the rank of the noncentrality parameter is constrained by the size of the shape parameter. This rank condition has recently been proved with different methods for the special case of symmetric, positive semidefinite matrices (Letac and Massam; Graczyk, Malecki, and Mayerhofer).