QUADRATIC FORMS UNDER SPHERICAL DISTRIBUTIONS

Let \undersetN \times PX=\binomX_(1)X_(2) be a spherically Symmetric distributed matrix, we shall prove the equivalence of the following propositions: 1. X₍₁₎andX₍₂₎are mutually independent; 2. X'₍₁₎X₍₁₎andX'₍₂₎X₍₂₎ are mutually independent; 3. Vec X is distributed as N(0, \nabla \otimes I) for some non-negative matrix V. At last, under the ristriction P（X=0）＜1 we extend the classical Cochran’s theorem of quadratic formt to a more general fashion.