Multivariate Versions of Cochran's Theorems II

A general easily checkable Cochran theorem is obtained for a normal random operator Y. This result does not require that the covariance, [Sigma]Y, of Y is nonsingular or is of the usual form A [circle times operator] [Sigma] ; nor does it assume that the mean, [mu], of Y is equal to zero. Indeed, {Y'WiY} (with nonnegative definite Wi's) is a family of independent Wishart random operators Y'WiY of parameter (mi, [Sigma], [lambda]i) if and only if for some nonnegative definite A and for all i [not equal to] j: (a)(Wi [circle times operator] I)([Sigma]Y - A [circle times operator] [Sigma])(Wi [circle times operator] I) = 0; (b) AWiAWi = AWi, r(AWi) = mi, (c) [lambda]i = [mu]'Wi[mu] = [mu]'WiAWi[mu]; and (d) (Wi [circle times operator] I)[Sigma]Y(Wj [circle times operator] I) = 0. The usual multivariate versions of Cochran's theorem are contained in a special case of our result where [Sigma]Y = A [circle times operator] [Sigma]. The A in our version of Cochran's theorem can actually be constructed from [Sigma], [Sigma]Y, and the sum of the Wi's.