796 CHAPTER 38. PROBABILITY

Corollary 38.9.9 Suppose X =(

X1 · · · Xn

)Twhere X has a moment generating

function of the formM (t) = e

12 t

T At

where A is real and symmetric having rank r ≤ n and eigenvalues 0 or 1. Then XT AX isX 2 (r). (When r < n, this is a moment generating function of a random variable which issaid to be a singular multivariate normal. )

Proof: By Theorem 11.4.7 there is orthogonal U such that UT AU = D where D is of

the form

(I 00 0

)where I is an r× r identity matrix. Then let Y =UTX. What is the

distribution of Y ?

E (exp(t ·Y )) ≡ E(exp(t·UTX

))= E (exp(Ut ·X))

= exp(

12(Ut)T A(Ut)

)= exp

(−1

2tTUT AUt

)= exp

(12tT Dt

)=

r

∏k=1

exp(

12

t2k

)Now exp(tYk) = 1 and so Yk = 0 if k > r and otherwise, Yk is n(0,1) , normal with mean 0and variance 1. Thus Theorem 38.9.7 implies that these random variables are independentand each n(0,1). Hence by Proposition 38.9.8,

XT AX = Y TUT AUY = Y T DY =r

∑k=1

Y 2k which is X 2 (r) . ■