2006 CHAPTER 61. PROBABILITY IN INFINITE DIMENSIONS

It follows there exists a probability measure µ defined on σ (E ) and random variables,Xy for each y ∈ Λ such that whenever

{y1, · · · ,yp

}⊆ Λ,

µ{y1,··· ,yp} (Ay1 ×·· ·×Ayn) = µ

(∏y∈Λ

Ay

)

where Ay = R whenever y /∈{

y1, · · · ,yp}

. This defines a measure on σ (E ) which consistsof sets of RΛ.

By Lemma 61.5.1 it follows{

θ−1 (U) : U ∈ σ (E )

}= σ (A ) which equals σ (X) . De-

fine ν on σ (X) byν (F)≡ µ (θF) .

Thus ν is a measure because µ is and θ is one to one.I need to check whether ν works. Let x = ∑

mk=1 tkyk and let a typical element of RΛ be

denoted by z. Then by Kolmogorov’s theorem above,∫X∗

exp

(ix∗(

m

∑k=1

tkyk

))dν =

∫X∗

exp

(i

(m

∑k=1

tkx∗ (yk)

))dν

=∫

X∗exp

(i

(m

∑k=1

tkπyk (θx∗)

))dν =

∫RΛ

exp

(i

(m

∑k=1

tkπyk z

))dµ

=∫Rm

exp(i(t ·x))dµ{y1,··· ,ym} (x) = ψ

(m

∑k=1

tkyk

)where the last equality comes from 61.5.16. Since Λ is a Hamel basis, it follows that forevery x ∈ X ,

ψ (x) =∫

X∗exp(ix∗ (x))dν

This proves the theorem.

61.6 The Multivariate Normal DistributionHere I give a review of the main theorems and definitions about multivariate normal randomvariables. Recall that for a random vector (variable), X having values in Rp, λ X is the lawof X defined by

P([X ∈ E]) = λ X (E)

for all E a Borel set inRp. In different notaion, L (X)= λ X. Then the following definitionsand theorems are proved and presented starting on Page 1910

Definition 61.6.1 A random vector, X, with values in Rp has a multivariate normal distri-bution written as X∼Np (m,Σ) if for all Borel E ⊆ Rp,

λ X (E) =∫Rp

XE (x)1

(2π)p/2 det(Σ)1/2 e−12 (x−m)∗Σ−1(x−m)dx

for µ a given vector and Σ a given positive definite symmetric matrix.