2180 CHAPTER 64. WIENER PROCESSES

measurable functions, ξ k : Ω→ R such that

P([

ξ i1 ∈ Fi1

]∩[ξ i2 ∈ Fi2

]· · ·∩

[ξ in ∈ Fin

])= µ i1···in (F1×·· ·×Fn)

= P([

ξ i1 ∈ Fi1

])· · ·P

([ξ in ∈ Fin

])which shows the random variables are independent as well as normal with mean 0 andvariance 1.

Recall that the sum of independent normal random variables is normal. The Wienerprocess is just an infinite weighted sum of the above independent normal random variables,the weights depending on t. Therefore, if the sum converges, it is not too surprising thatthe result will be normally distributed and the variance will depend on t. This is the ideabehind the following theorem.

Theorem 64.1.3 There exists a real Wiener process as defined in Definition 64.1.1. Fur-thermore, the distribution of W (t)−W (s) is the same as the distribution of W (t− s) andW is Holder continuous with exponent γ for any γ < 1/2. Also for each α > 1,

E(|W (t)−W (s)|α

)≤Cα |t− s|α/2 E

(|W (1)|α

)Proof: Let {gm}∞

m=1 be a complete orthonormal set in L2 (0,∞) . Thus, if f ∈ L2 (0,∞) ,

f =∞

∑i=1

( f ,gi)L2 gi.

The Wiener process is defined as

W (t,ω)≡∞

∑i=1

(X(0,t),gi

)L2 ξ i (ω)

where the random variables, {ξ i} are as described in Lemma 64.1.2. The series convergesin L2 (Ω) where (Ω,F ,P) is the probability space on which the random variables, ξ i aredefined. This will first be shown. Note first that from the independence of the ξ i,∫

Ω

ξ iξ jdP = 0

Therefore,

∫Ω

∣∣∣∣∣ n

∑i=m

(X(0,t),gi

)L2 ξ i (ω)

∣∣∣∣∣2

dP =n

∑i=m

(X(0,t),gi

)2L2

∫Ω

|ξ i|2 dP

=n

∑i=m

(X(0,t),gi

)2L2

which converges to 0 as m,n → ∞. Thus the partial sums are a Cauchy sequence inL2 (Ω,P) .