64.4. INDEPENDENT INCREMENTS AND MARTINGALES 2189

= limn→∞

n

∏k=1

E (exp(iφ (ek)(ψk (t)−ψk (s))))

= limn→∞

n

∏k=1

E(

exp(−1

2φ (ek)

2 (t− s)))

= limn→∞

E

(exp

n

∑k=1

(−1

2φ (ek)

2 (t− s)))

which is the same as the result for

E (exp(iφ (W (t− s))))

andE(exp(iφ(√

t− sW (1))))

.

This has proved the following lemma.

Lemma 64.3.3 Let E be a real separable Banach space. Then there exists an E valuedstochastic process, W (t) such that L (W (t)) and L (W (t)−W (s)) are Gaussian mea-sures and the increments, {W (t)−W (s)} are independent. Furthermore, the incrementW (t)−W (s) has the same distribution as W (t− s) and W (t) has the same distribution as√

tW (1).

Now I want to consider the question of Holder continuity of the functions, t→W (t,ω).∫Ω

||W (t)−W (s)||α dP =∫

E||x||α dµW (t)−W (s)

=∫

E||x||α dµW (t−s) =

∫E||x||α dµ√t−sW (1)

=∫

Ω

∣∣∣∣√t− sW (1)∣∣∣∣α dP

= |t− s|α/2∫

Ω

||W (1)||α dP =Cα |t− s|α/2

by Fernique’s theorem, Theorem 61.7.5. From the Kolmogorov Čentsov theorem, Theorem62.2.2, it follows {W (t)} is Holder continuous with exponent γ <

(α

2 −1)/α.

This completes the proof of the following theorem.

Theorem 64.3.4 Let E be a separable real Banach space. Then there exists a stochas-tic process, {W (t)} such that the distribution of W (t) and every increment, W (t)−W (s)is Gaussian. Furthermore, the increments corresponding to disjoint intervals are indepen-dent, L (W (t)−W (s)) =L (W (t− s)) =L

(√t− sW (1)

). Also for a.e. ω, t→W (t,ω)

is Holder continuous with exponent γ < 1/2.

64.4 Independent Increments and MartingalesHere is an interesting lemma.