Cochran's Theorem - Statement

Statement

Suppose U₁, ..., U_n are independent standard normally distributed random variables, and an identity of the form

$\sum_{i=1}^n U_i^2=Q_1+\cdots + Q_k$

can be written, where each Q_i is a sum of squares of linear combinations of the Us. Further suppose that

$r_1+\cdots +r_k=n$

where r_i is the rank of Q_i. Cochran's theorem states that the Q_i are independent, and each Q_i has a chi-squared distribution with r_i degrees of freedom.

Here the rank of Q_i should be interpreted as meaning the rank of the matrix B(i), with elements B_j,k(i), in the representation of Q_i as a quadratic form:

Less formally, it is the number of linear combinations included in the sum of squares defining Q_i, provided that these linear combinations are linearly independent.

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