64.1. REAL WIENER PROCESSES 2181
It just remains to verify this definition satisfies the desired conditions. First I will showthat ω →W (t,ω) is normally distributed with mean 0 and variance t. That it should benormally distributed is not surprising since it is just a sum of independent random variableswhich are this way. Selecting a suitable subsequence, {nk} it can be assumed
W (t,ω) = limk→∞
nk
∑i=1
(X(0,t),gi
)L2 ξ i (ω) a.e.
and so from the dominated convergence theorem and the independence of the ξ i,
E (exp(iλW (t))) = limk→∞
E
(exp
(iλ
nk
∑j=1
(X(0,t),g j
)L2 ξ j (ω)
))
= limk→∞
E
(nk
∏j=1
exp(
iλ(X(0,t),g j
)L2 ξ j (ω)
))
= limk→∞
nk
∏j=1
E(
exp(
iλ(X(0,t),g j
)L2 ξ j (ω)
))= lim
k→∞
nk
∏j=1
e−12 λ
2(X(0,t),g j)2L2
= limk→∞
exp
(nk
∑j=1−1
2λ
2 (X(0,t),g j)2
L2
)
= exp(−1
2λ
2 ∣∣∣∣X(0,t)∣∣∣∣2
L2
)= exp
(−1
2λ
2t),
the characteristic function of a normally distributed random variable having variance t andmean 0.
It is clear W (0) = 0. It remains to verify the increments are independent. To do this,consider
E (exp(i [λ (W (t)−W (s))+µ (W (s)−W (r))])) (64.1.1)
Is this equal to
E (exp(i [λ (W (t)−W (s))]))E (exp(i [µ (W (s)−W (r))]))? (64.1.2)
Letting nk→ ∞ such that convergence happens pointwise for each function of interest, andusing the independence of the ξ i, and the dominated convergence theorem as needed,
E
(exp
(i
[∞
∑i=1
λ(X(s,t),gi
)L2 ξ i +
∞
∑i=1
µ(X(r,s),gi
)L2 ξ i
]))
= limk→∞
E
(exp
(i
[nk
∑j=1
(λ(X(s,t),g j
)L2 +µ
(X(r,s),g j
)L2
)ξ j
]))