28.4. PROKHOROV AND LEVY THEOREMS 775

and so, doing this repeatedly, one obtains for the above integral an expression of the form

2p pm−1

(2π)p/2 Cm

(p

∏k=1

σ k

)m

= 2p pm−1

(2π)p/2 Cm det(Σ)m

which shows that the moments of X all exist and are dominated by an expression whichdepends continuously on det(Σ) . ■

In particular, these moments are bounded in case Σn → Σ where perhaps det(Σ) = 0but Σn is positive definite. With Corollary 28.4.11, this has proved the following theoremabout the generalized normal distribution.

Theorem 28.4.14 Let Σ be nonnegative and self adjoint p× p matrix. Then thereexists a random variable X whose distribution measure λX has characteristic functionψ (t)≡ e−

12 t∗Σt. Then all the moments exist and E (XX∗) = Σ.

Proof: It remains to verify E (XX∗) = Σ but this is routine from the fact that themoments exist. Use the characteristic function to compute E (XiX j). Take d

dt j

(d

dti(ψ (t))

).

Using repeated index summation convention,

ψ (t) = e−12 trΣrsts ,ψ ti = e−

12 trΣrsts (−Σists) ,ψ tit j

= e−12 trΣrsts (−Σ jsts)(−Σists)+ e−

12 trΣrsts (−Σi j)

Thus i2E (XiX j) =−Σi j showing that E (XX∗) = Σ as claimed. ■The case where m= 0 is the one of most interest here, but you could always reduce

to this case by considering a random variable X−m where E (X) =m. There is aninteresting corollary to this theorem.

Corollary 28.4.15 Let H be a real Hilbert space. Then there exist random variablesW (h) for h ∈ H such that for any finite set { f1, f2, · · · , fn},

(W ( f1) ,W ( f2) , · · · ,W ( fn))

is normally distributed with mean 0 and covariance Σi j = ( fi, f j) and for every h,g,

E (W (h)W (g)) = (h,g)H

If {ei} is an orthogonal set of vectors of H, then {W (ei)} are independent random vari-ables.

Proof: Let µh1···hmbe a generalized multivariate normal probability distribution with

covariance Σi j = (hi,h j) and mean 0. That such a thing exists follows from Theorem28.4.14. Thus the characteristic function of this probability measure is e−

12 t∗Σt. Now con-

sider Ek1 ×·· ·×Ekn for Borel sets Ek j where {h1, · · · ,hm} ⊆ {k1 · · ·kn} for n > m and theset Ek j = R whenever k j /∈ {h1, · · · ,hm} For simplicity, say h1, · · · ,hm are the first m slotsof k1, · · · ,kn. Now consider µk1···kn

,

{h1 · · ·hm,km+1 · · ·kn}= {k1 · · ·kn}

Let ν be a measure on B (Rm) which is given by ν (E) ≡ µk1···kn(E×Rn−m). Then does

it follow that ν = µh1···hm? If so, then the Kolmogorov consistency condition will hold for