Characterization of Wishart-Laplace distributions via Jordan algebra homomorphisms

摘要

For a real, Hermitian, or quaternion normal random matrix Y with mean zero, necessary and sufficient conditions for a quadratic form Q(Y) to have a Wishart-Laplace distribution (the distribution of the difference of two independent central Wishart W-rho(m(i), Sigma) random matrices) are given in terms of a certain Jordan algebra homomorphism rho. Further, it is shown that {Q(k)(Y)} is independent Laplace-Wishart if and only if in addition to the aforementioned conditions, the images rho(k)(Sigma(+)) of the Moore-Penrose inverse Sigma(+) of Sigma are mutually orthogonal: rho(k)(Sigma(+))rho(l)(Sigma(+)) = 0 for k not equal l.