If X = X 1 ,, X p are jointly normal with zero mean and co-variance matrix i,j of full rank, show that the

If X = X 1

,…, X p

are jointly normal with zero mean and co-variance matrix κ

i,j of full rank, show that the rth order cumulant of Y =κi,jX iX j

is 2 r−1 p(r − 1)!.

Hence show that the cumulant generating function of Y is −

1 2 p log (1 − 2ξ) and therefore that Y has the χ

2 distribution on p degrees of freedom.