1898 CHAPTER 59. BASIC PROBABILITY

59.13 Characteristic Functions In Banach SpaceI will consider the relation between the characteristic function and independence of randomvariables having values in a Banach space. Recall an earlier proposition which relatesindependence of random vectors with characteristic functions. It is proved starting on Page1891.

Proposition 59.13.1 Let {Xk}nk=1be random vectors such that Xk has values in Rpk . Then

the random vectors are independent if and only if

E(eiP)= n

∏j=1

E(eit j ·X j

)where P≡ ∑

nj=1 t j ·X j for t j ∈ Rp j .

It turns out there is a generalization of the above proposition to the case where therandom variables have values in a real separable Banach space. Before proving this recallan earlier theorem which had to do with reducing to the case where the random variableshad values in Rn, Theorem 59.5.1. It is restated here for convenience.

Theorem 59.13.2 The random variables {Xi}i∈I are independent if whenever

{i1, · · · , in} ⊆ I,

mi1 , · · · ,min are positive integers, and gmi1, · · · ,gmin

are in(E ′)mi1 , · · · ,

(E ′)min

respectively,{

gmi j◦Xi j

}n

j=1are independent random vectors having values in

Rmi1 , · · · ,Rmin

respectively.

Now here is the theorem about independence and the characteristic functions.

Theorem 59.13.3 Let {Xk}nk=1be random variables such that Xk has values in Ek, a real

separable Banach space. Then the random variables are independent if and only if

E(eiP)= n

∏j=1

E(

eit∗j (X j))

where P≡ ∑nj=1 t∗j (X j) for t∗j ∈ E ′j.

Proof: If the random variables are independent, then so are the random variables,t∗j (X j) and so the equation follows.

The interesting case is when the equation holds.