Cochran’s theorem
Let X be multivariate normally distributed as ${\bm{N}}_{\bm{p}}\mathbf{(}\mathrm{\U0001d7ce}\mathbf{,}\bm{I}\mathbf{)}$ such that
$${\text{\mathbf{X}}}^{\mathrm{T}}\text{\mathbf{X}}=\sum _{i=1}^{k}{Q}_{i},$$ 
where each

1.
${Q}_{i}$ is a quadratic form^{}

2.
${Q}_{i}={\text{\mathbf{X}}}^{\mathrm{T}}{\text{\mathbf{B}}}_{i}\text{\mathbf{X}}$, where ${\text{\mathbf{B}}}_{i}$ is a $p$ by $p$ square matrix^{}

3.
${\text{\mathbf{B}}}_{i}$ is positive semidefinite^{}

4.
$\mathrm{rank}({\text{\mathbf{B}}}_{i})={r}_{i}$
Then any two of the following imply the third:

1.
${\sum}_{i=1}^{k}{r}_{i}=p$

2.
each ${Q}_{i}$ has a chi square distribution (http://planetmath.org/ChiSquaredRandomVariable) with ${r}_{i}$ of freedom, ${\chi}^{2}({r}_{i})$

3.
${Q}_{i}$’s are mutually independent
As an example, suppose $X_{1}{}^{2}\sim {\chi}^{2}({m}_{1})$ and $X_{2}{}^{2}\sim {\chi}^{2}({m}_{2})$. Furthermore, assume $X_{1}{}^{2}\ge X_{2}{}^{2}$ and ${m}_{1}>{m}_{2}$, then
$$X_{1}{}^{2}X_{2}{}^{2}\sim {\chi}^{2}({m}_{1}{m}_{2}).$$ 
This corollary is known as Fisher’s theorem.
Title  Cochran’s theorem 

Canonical name  CochransTheorem 
Date of creation  20130322 14:33:01 
Last modified on  20130322 14:33:01 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  8 
Author  CWoo (3771) 
Entry type  Theorem 
Classification  msc 62J10 
Classification  msc 62H10 
Classification  msc 62E10 
Defines  Fisher’s theorem 