780 CHAPTER 28. THE NORMAL DISTRIBUTION

is a set of λZ continuity due to the assumption that λZ ≪ mp which is implied by Z ∼Np (0,Σ). ■

Suppose X is a random vector with covariance Σ and mean m, and suppose also thatΣ−1 exists. Consider Σ−(1/2) (X−m)≡ Y. Then E (Y ) = 0 and

E (Y Y ∗) = E(

Σ−(1/2) (X−m)(X∗−m)Σ

−(1/2))

= Σ−(1/2)E ((X−m)(X∗−m))Σ

−(1/2) = I.

Thus Y has zero mean and covariance I. This implies the following corollary to Theorem28.5.8.

Corollary 28.5.9 Let{X j}∞

j=1 be independent identically distributed random vari-

ables and suppose they have meanm and positive definite covariance Σ where Σ−1 exists.Then if

Zn ≡n

∑j=1

Σ−(1/2) (X j−m)√n

,

it follows that for Z ∼ Np (0,I), FZn (x)→ FZ (x) for all x.