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

Abstract

Craig's theorem concerns conditions necessary and sufficient for the stochastic independence of two quadratic forms in variates following a joint normal distribution. We give a standard formulation of Craig's theorem, one that encompasses both the nonsingular and the singular cases. But we prove this result using only linear algebra and calculus.

Key Words:

• Cumulants
• Independence
• Normal variates
• Quadratic forms