64.4. INDEPENDENT INCREMENTS AND MARTINGALES 2193

Theorem 64.4.3 Let {W (t)} be a Wiener process having values in a separable Banachspace as described in Theorem 64.3.4. There exists a set of measure 0, N such that forω /∈ N, the sum in 64.3.11 converges uniformly to W (t,ω) on any interval, [0,T ] . That is,for each ω not in a set of measure zero, the partial sums of the sum in that formula convergeuniformly to t→W (t,ω) on [0,T ].

Proof: By Lemma 64.4.2 the independence of the increments imply

n

∑k=m

ψk (t)ek

is a martingale and so by Theorem 62.5.3,

P

([sup

t∈[0,T ]

∣∣∣∣∣∣∣∣∣∣ n

∑k=m

ψk (t)ek

∣∣∣∣∣∣∣∣∣∣≥ α

])≤ 1

α

∫Ω

∣∣∣∣∣∣∣∣∣∣ n

∑k=m

ψk (T )ek

∣∣∣∣∣∣∣∣∣∣dP

From Corollary 64.3.2

∫Ω

∣∣∣∣∣∣∣∣∣∣ n

∑k=m

ψk (T )ek

∣∣∣∣∣∣∣∣∣∣dP ≤

n

∑k=m

∫Ω

|ψk (T )|dPλ k

≤n

∑k=m

λ k

which shows that there exists a subsequence, ml such that whenever n > ml ,

P

([sup

t∈[0,T ]

∣∣∣∣∣∣∣∣∣∣ n

∑k=ml

ψk (t)ek

∣∣∣∣∣∣∣∣∣∣≥ 2−k

])≤ 2−k.

Recall Lemma 59.15.6 stated below for convenience.

Lemma 64.4.4 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. (64.4.16)

Then in fact,Sk (ω)→ S (ω) a.e.ω (64.4.17)

Apply this lemma to the situation in which the Banach space, E is C ([0,T ] ;E) andζ k = ψkek. Then you can conclude uniform convergence of the partial sums,

m

∑k=1

ψk (t)ek.

This proves the theorem.