752 CHAPTER 27. ANALYTICAL CONSIDERATIONS

it follows

E(eiP)= n

∏k=1

E(

eitmk ·(gmk◦Xk

)). (27.10)

However, the expression on the right in 27.10 equals

n

∏k=1

E(

ei(tmk ·gmk

)◦Xk

)and tmk ·gmk

≡ ∑mkj=1 t jx∗j ∈ E ′. Also the expression on the left equals

E(

ei∑nk=1 tmk ·gmk

◦Xk)

Therefore, by assumption, 27.10 holds. ■There is an obvious corollary which is useful.

Corollary 27.6.4 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 ∈M j where M j is a dense subset of E ′j.

Proof: The easy direction follows from Theorem 27.6.3. Suppose then the above equa-tion holds for all t∗j ∈ M j. Then let t∗j ∈ E ′ and let

{t∗n j

}be a sequence in M j such that

limn→∞ t∗n j = t∗j in E ′. Then define

P≡n

∑j=1

t∗j X j, Pn ≡n

∑j=1

t∗n jX j.

It follows

E(eiP)= lim

n→∞E(eiPn)= lim

n→∞

n

∏j=1

E(

eit∗n j(X j))=

n

∏j=1

E(

eit∗j (X j))■

27.7 Convolution and SumsLemma 26.1.9 on Page 717 makes possible a definition of convolution of two probabilitymeasures defined on B (E) where E is a separable Banach space. I will first show a littletheorem about density of continuous functions in Lp (E) and then define the convolution oftwo finite measures. First here is a simple technical lemma.

Lemma 27.7.1 Suppose K is a compact subset of U an open set in E a metric space.Then there exists δ > 0 such that

dist(x,K)+dist(x,UC)≥ δ for all x ∈ E.