Abstract
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. 868â874] 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 setâas is the case for Riesz measuresâbut 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).
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