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, ΣY, of Y is nonsingular or is of the usual form A ⊗ Σ ; nor does it assume that the mean, μ, 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, Σ, λi) if and only if for some nonnegative definite A and for all i ≠ j: (a)(Wi ⊗ I)(ΣY − A ⊗ Σ)(Wi ⊗ I) = 0; (b) AWiAWi = AWi, r(AWi) = mi, (c) λi = μ′Wiμ = μ′WiAWiμ; and (d) (Wi ⊗ I)ΣY(Wj ⊗ I) = 0. The usual multivariate versions of Cochran′s theorem are contained in a special case of our result where ΣY = A ⊗ Σ. The A in our version of Cochran′s theorem can actually be constructed from Σ, ΣY, and the sum of the Wi′s.