ON TRANSFORMATIONS AND DETERMINANTS OF WISHART VARIABLES ON SYMMETRICAL CONES

Let x and y be independent Wishart random variables on a simple Jordan algebra V. If c is a given idempotent of V, write x = x(1) + x(12) x(0) for the decomposition of x in V(c, 1) + V(c, 1/2) + V(c, 0) where V(c, lambda) equals the set of v such that cv = lambda v. In this pap er we compute E(det(ax + by)) and some generalizations of it (Theorems 5 and 6). We give the joint distribution of (x(1), x(12), y(0)) where y(0) = x(0) - P(x(12)) x(1)(-1) and P is the quadratic representation in V. In statistics, if x is a real positive definite matrix divided into the blocks x(11), x(12), x(21), x(22), then y(o) is equal to x(22 .1) = x(22) - x(21)x(11)(-1)x(12). We also compute the joint distribut ion of the eigenvalues of x (Theorem 9). These results have been known only when V is the algebra of Hermitian matrices with entries in the real or the complex field. To obtain our results, we need to prove sev eral new results on determinants in Jordan algebras. They include in p articular extensions of some classical parts of linear algebra like Le ibnitz's determinant formula (Proposition 2) or Schur's complement (Eq s. (3.3) and (3.6)).