2188 CHAPTER 64. WIENER PROCESSES

= limn→∞

E

(m

∏j=1

exp

(i

n

∑k=1

(ψk (t j)−ψk

(t j−1

))φ j (ek)

))

= limn→∞

m

∏j=1

n

∏k=1

E(

exp(

i(ψk (t j)−ψk

(t j−1

))φ j (ek)

))= lim

n→∞

m

∏j=1

E

(exp

(i

n

∑k=1

(ψk (t j)−ψk

(t j−1

))φ j (ek)

))

= limn→∞

E

(m

∏j=1

exp

(i

n

∑k=1

(ψk (t j)−ψk

(t j−1

))φ j (ek)

))

= limn→∞

m

∏j=1

n

∏k=1

E(

exp(

i(ψk (t j)−ψk

(t j−1

))φ j (ek)

))= lim

n→∞

m

∏j=1

E

(exp

(i

n

∑k=1

(ψk (t j)−ψk

(t j−1

))φ j (ek)

))

=m

∏j=1

E

(exp

(i

∞

∑k=1

(ψk (t j)−ψk

(t j−1

))φ j (ek)

))

=m

∏j=1

E

(exp

(iφ j

(∞

∑k=1

(ψk (t j)−ψk

(t j−1

))ek

)))

=m

∏j=1

E(

exp(

iφ j(W (t j)−W

(t j−1

))))which shows by Theorem 59.13.3 on Page 1898 that the random vectors,{

W (t j)−W(t j−1

)}mj=1

are independent.It is also routine to verify using properties of the ψk and characteristic functions that

L (W (t)−W (s)) = L (W (t− s)). To see this, let φ ∈ E ′

E (exp(iφ (W (t)−W (s))))

= E

((exp

(iφ

∞

∑k=1

(ψk (t)−ψk (s))ek

)))

= limn→∞

E

((exp

(iφ

n

∑k=1

(ψk (t)−ψk (s))ek

)))