2182 CHAPTER 64. WIENER PROCESSES

= limk→∞

E

(nk

∏j=1

exp(

i(

λ(X(s,t),g j

)L2 +µ

(X(r,s),g j

)L2

)ξ j

))

= limk→∞

nk

∏j=1

E(

exp(

i(

λ(X(s,t),g j

)L2 +µ

(X(r,s),g j

)L2

)ξ j

))

= limk→∞

nk

∏j=1

exp(−1

2(λX(s,t)+µX(r,s),g j

)2L2

)

= limk→∞

exp

(−1

2

nk

∑j=1

(λX(s,t)+µX(r,s),g j

)2L2

)

= exp

(−1

2

∞

∑j=1

(λX(s,t)+µX(r,s),g j

)2L2

)= exp

(−1

2

∥∥λX(s,t)+µX(r,s)∥∥2

L2

)

= exp(−1

2

[λ

2∥∥X(s,t)∥∥2

L2 +µ2∥∥X(r,s)

∥∥2L2

])because the functions λX(s,t),µX(r,s) are orthogonal. Then this equals

= exp(−1

2

[λ

2 (t− s)+µ2 (s− r)

])

= exp(−1

2(t− s)λ

2)

exp(−1

2(s− r)µ

2)

which equals 64.1.2 and this shows the increments are independent. Obviously, this sameargument shows this holds for any finite set of disjoint increments.

From the definition, if t > s

W (t− s) =∞

∑k=1

(X(0,t−s),gk

)L2 ξ k

while

W (t)−W (s) =∞

∑k=1

(X(s,t),gk

)L2 ξ k.

Then the same argument given above involving the characteristic function to show W (t)is normally distributed shows both of these random variables are normally distributed withmean 0 and variance t− s because they have the same characteristic function.