61.6. THE MULTIVARIATE NORMAL DISTRIBUTION 2007

Theorem 61.6.2 For X∼ Np (m,Σ) ,m = E (X) and

Σ = E((X−m)(X−m)∗

).

Theorem 61.6.3 Suppose X1∼Np (m1,Σ1) , X2∼Np (m2,Σ2) and the two random vectorsare independent. Then

X1 +X2 ∼ Np (m1 +m2,Σ1 +Σ2). (61.6.17)

Also, if X∼ Np (m,Σ) then −X∼ Np (−m,Σ) . Furthermore, if X∼ Np (m,Σ) then

E(eit·X)= eit·me−

12 t∗Σt (61.6.18)

Also if a is a constant and X∼ Np (m,Σ) then aX∼ Np(am,a2Σ

).

Following [103] a random vector has a generalized normal distribution if its character-istic function is given as

eit·me−12 t∗Σt (61.6.19)

where Σ is symmetric and has nonnegative eigenvalues. For a random real valued vari-able, m is scalar and so is Σ so the characteristic function of such a generalized normallydistributed random variable is

eitme−12 t2σ2

(61.6.20)

These generalized normal distributions do not require Σ to be invertible, only that the eigen-values be nonnegative. In one dimension this would correspond the characteristic functionof a dirac measure having point mass 1 at m. In higher dimensions, it could be a mixtureof such things with more familiar things. I will often not bother to distinguish betweengeneralized normal and normal distributions.

Here are some other interesting results about normal distributions found in [103]. Thenext theorem has to do with the question whether a random vector is normally distributedin the above generalized sense. It is proved on Page 1913.

Theorem 61.6.4 Let X = (X1, · · · ,Xp) where each Xi is a real valued random variable.Then X is normally distributed in the above generalized sense if and only if every linearcombination, ∑

pj=1 aiXi is normally distributed. In this case the mean of X is

m = (E (X1) , · · · ,E (Xp))

and the covariance matrix for X is

Σ jk = E ((X j−m j)(Xk−mk))

where m j = E (X j).

Also proved there is the interesting corollary listed next.

Corollary 61.6.5 Let X = (X1, · · · ,Xp) ,Y = (Y1, · · · ,Yp) where each Xi,Yi is a real valuedrandom variable. Suppose also that for every a ∈ Rp, a ·X and a ·Y are both normallydistributed with the same mean and variance. Then X and Y are both multivariate normalrandom vectors with the same mean and variance.