CRAIG-SAKAMOTO’S THEOREM FOR THE WISHART DISTRIBUTIONS ON SYMMETRIC CONES

G. LETAC^{1} AND H. MASSAM^{2}

1Laboratoire de Statistique et Probabilités, Université Paul Sabatier, 31062 Toulouse, France

2Department of Mathematics and Statistics, York University, North York, Ontario, Canada M3J 1P3

(Received May 27, 1994; revised January 30, 1995)

Abstract. A version of Craig-Sakamoto’s theorem says essentially that if X is a N(0, I_{n}) Gaussian random variable in R^{n}, and if A and B are (n, n) symmetric matrices, then X′AX and X′BX (or traces of AXX′ and BXX′) are independent random variables if and only if AB = 0. As observed in 1951, by Ogasawara and Takahashi, this result can be extended to the case where XX′ is replaced by a Wishart random variable. Many properties of the ordinary Wishart distributions have recently been extended to the Wishart distributions on the symmetric cone generated by a Euclidean Jordan algebra E. Similarly, we generalize there the version of Craig’s theorem given by Ogasawara and Takahashi. We prove that if a and b are in E and if W is Wishart distributed, then Trace a.W and Trace b.W are independent if and only if a.b = 0 and a.(b.x) = b.(a.x) for all x in E, where the . indicates Jordan product.

Key words and phrases: Jordan algebra, Wishart distributions, exponential families on convex cones.