64.5. HILBERT SPACE VALUED WIENER PROCESSES 2207

Therefore, eiP equals the expression in 64.5.32 because both equal the expression in 64.5.31and it follows from Proposition 59.11.1 on Page 1891 that the random variables of 64.5.29are independent.

What about the claim of uniform convergence? By the independence of the increments,it follows from Lemma 64.4.2 that {W (t)} is a martingale and each real valued function,(W (t) ,ek)U is also a martingale. Therefore, Theorem 62.5.3 can be applied to conclude

P

([sup

t∈[0,T ]

∣∣∣∣∣ n

∑k=m

(W (t) ,ek)U ek

∣∣∣∣∣≥ α

])≤ 1

α

∫Ω

∣∣∣∣∣ n

∑k=m

(W (T ) ,ek)U ek

∣∣∣∣∣dP

≤ 1α

∫Ω

∣∣∣∣∣ n

∑k=m

(W (T ) ,ek)U ek

∣∣∣∣∣2

dP =1α

n

∑k=m

∫Ω

(W (T ) ,ek)2U dP

=1α

n

∑k=m

(Qek,ek)T =Tα

n

∑k=m

λ k ≤Tα

∞

∑k=m

λ k

Since ∑∞k=1 λ k < ∞, there exists a sequence, {ml} such that if n > ml

P

([sup

t∈[0,T ]

∣∣∣∣∣ n

∑k=ml

(W (t) ,ek)U ek

∣∣∣∣∣> 2−k

])< 2−k

and so by the Borel Cantelli lemma, off a set of measure 0 the partial sums{ml

∑k=1

(W (t) ,ek)U ek

}converge uniformly on [0,T ] . This is very interesting but more can be said. In fact theoriginal partial sums converge.

Recall Lemma 59.15.6 stated below for convenience.

Lemma 64.5.5 Let {ζ k} be a sequence of random variables having values in a separablereal Banach space, E whose distributions are symmetric. Letting Sk ≡ ∑

ki=1 ζ i, suppose{

Snk

}converges a.e. Also suppose that for every m > nk,

P([∣∣∣∣Sm−Snk

∣∣∣∣E > 2−k

])< 2−k. (64.5.33)

Then in fact,Sk (ω)→ S (ω) a.e.ω (64.5.34)

Apply this lemma to the situation in which the Banach space, E is C ([0,T ] ;U) . Thenyou can conclude uniform convergence of the partial sums,

m

∑k=1

(W (t) ,ek)U ek.

This proves the theorem.