Let be a column vector of independent N(0,1) distributed random variables. Let A be an orthogonal matrix.
Prove that using moment generating functions.
Let and
If they are independent, then the MGF of the sum is the product of the individual MGFs.
So
Thus
NOW, let which shows that
the sum of any two independent s
is a , you just add the degrees of freedom.
This can easily be extended to n random variables.