An Accessible Proof of Craig's Theorem in the Noncentral Case

Abstract

Abstract Craig's theorem gives a simple, necessary and sufficient condition for the stochastic independence of two quadratic forms in variates following an arbitrary nonsingular joint normal distribution. Previous correct proofs in the noncentral case (i.e., some means nonzero) have required use of advanced mathematical theory. We give a proof that requires only linear algebra and calculus, making the noncentral case of the theorem as accessible as the central case. Furthermore, our method of proof shows that, in proving necessity, the assumption of independence can be weakened to conditions involving a finite number of the joint moments of the quadratic forms.