Wiener measure and Donsker’s theorem

Jordan Bell
September 4, 2015

1 Relatively compact sets of Borel probability measures on C[0,1]

Let E=C[0,1], let E be the Borel σ-algebra of E, and let 𝒫E be the collection of Borel probability measures on E. We assign 𝒫 the narrow topology, the coarsest topology on 𝒫E such that for each FCb(E) the map μEF𝑑μ is continuous.

For fE and δ>0 we define


For fE, ωf(δ)0 as δ0, and for δ>0, fωf(δ) is continuous. We shall use the following characterization of a relatively compact subset A of E, which is proved using the Arzelà-Ascoli theorem.

Lemma 1.

Let A be a subset of E. A¯ is compact if and only if




We shall use Prokhorov’s theorem:11 1 K. R. Parthasarathy, Probability Measures on Metric Spaces, p. 47, Chapter II, Theorem 6.7. for X a Polish space and for Γ𝒫X, Γ¯ is compact if and only if for each ϵ>0 there is a compact subset Kϵ of X such that μ(Kϵ)1-ϵ for all μΓ. Namely, a subset of 𝒫X is relatively compact if and only if it is tight. We use Prokhorov’s theorem to prove a characterization of relatively compact subsets of 𝒫E, which we then use to prove the characterization in Theorem 3.22 2 K. R. Parthasarathy, Probability Measures on Metric Spaces, p. 213, Chapter VII, Lemma 2.2.

Lemma 2.

Let Γ be a subset of 𝒫E. Γ¯ is compact if and only if for each ϵ>0 there is some Mϵ< and a function δωϵ(δ) satisfying ωϵ(δ)0 as δ0 and such that for all μΓ,



Aϵ={fE:|f(0)|Mϵ},Bϵ={fE:ωf(δ)ωϵ(δ) for all δ>0}.

Suppose that Γ satisfies the above conditions. Because f|f(0)| is continuous, Aϵ is closed. For δ>0, suppose that fn is a sequence in Bϵ tending to some fE. Because gωg(δ) is continuous, ωfn(δ)ωf(δ), and because ωfn(δ)ωϵ(δ) for each n, we get ωf(δ)ωϵ(δ) and hence fBϵ, showing that Bϵ is closed. Therefore Kϵ=AϵBϵ is closed, i.e. Kϵ=Kϵ¯. The set Kϵ satisfies



lim supδ0supfKϵωf(δ)lim supδ0ωϵ(δ)=0,

thus by Lemma 1, Kϵ is compact. For μΓ,


and because Kϵ is compact, this means that Γ is tight, so by Prokhorov’s theorem, Γ is relatively compact.

Now suppose that Γ is relatively compact and let ϵ>0. By Prokhorov’s theorem, there is a compact set Kϵ in E such that μ(Kϵ)1-ϵ2 for all μΓ. Define


Because Kϵ is compact, by Lemma 1 we get that Mϵ< and ωϵ(δ)0 as δ0. For μΓ,


showing that Γ satisfies the conditions of the theorem. ∎

We now prove the characterization of relatively compact subsets of 𝒫E that we shall use in our proof of Donsker’s theorem.33 3 K. R. Parthasarathy, Probability Measures on Metric Spaces, p. 214, Chapter VII, Theorem 2.2.

Theorem 3 (Relatively compact sets in 𝒫).

Let Γ be a subset of 𝒫E. Γ¯ is compact if and only if the following conditions are satisfied:

  1. 1.

    For each ϵ>0 there is some Mϵ< such that

  2. 2.

    For each ϵ>0 and δ>0 there is some η=η(ϵ,δ)>0 such that


Suppose that Γ¯ is compact and let ϵ>0. By Lemma 2, there is some Mϵ< and a function ηωϵ(η) satisfying ωϵ(η)0 as η0 and


For δ>0, there is some η=η(ϵ,δ) with ωϵ(η)δ. Then for μΓ,


Now suppose that the conditions of the theorem hold. For each ϵ>0 and n1 there is some ηϵ,n>0 such that






for which


For fKϵ, then for each n1 we have fFϵ,n, which means that ωf(ηϵ,n)1n, and therefore


Thus for n1, if 0<ηηϵ,n then


which shows supfKϵωf(η)0 as η0. Then because


applying Lemma 1 we get that Kϵ¯ is compact. The map fωf(ηϵ,n) is continuous, so the set Fϵ,n is closed, and therefore the set Kϵ is closed. Because Kϵ is compact and μ(Kϵ)1-ϵ2 for all μΓ, it follows from by Prokhorov’s theorem that Γ is relatively compact. ∎

2 Wiener measure

For t1,,td[0,1], t1<<td, define πt1,,td:Ed by


which is continuous. We state the following results, which we will use later.44 4 K. R. Parthasarathy, Probability Measures on Metric Spaces, p. 212, Chapter VII, Theorem 2.1.

Theorem 4 (The Borel σ-algebra of E).

E is equal to the σ-algebra generated by {πt:t[0,1]}.

Two elements μ and ν of 𝒫E are equal if and only if for any d and any t1<<td, the pushforward measures


are equal.

Let (ξt)t[0,1] be a stochastic process with state space and sample space (Ω,,P). For t1<<td, let ξt1,,td=ξt1ξtd and let Pt1,,td=(ξt1,,td)*P: for Bd,


Pt1,,td is a Borel probability measure on d and is called a finite-dimensional distribution of the stochastic process.

The Kolmogorov continuity theorem55 5 K. R. Parthasarathy, Probability Measures on Metric Spaces, p. 216, Chapter VII, Theorem 3.1 tells us that if there are α,β,K>0 such that for all s,t[0,1],


then there is a unique μ𝒫E such that for all k and for all t1<<td,


We now define and prove the existence of Wiener measure.66 6 K. R. Parthasarathy, Probability Measures on Metric Spaces, p. 218, Chapter VII, Theorem 3.2.

Theorem 5 (Wiener measure).

There is a unique Borel probability measure W on E satisfying:

  1. 1.


  2. 2.

    For 0t0<t1<<td1 the random variables


    are independent (E,E,W)(,).

  3. 3.

    If 0s<t1, the random variable πt-πs:(E,E,W)(,) is normal with mean 0 and variance t-s.


There is a stochastic process (ξt)t[0,1] with state space and some sample space (Ω,,P), such that (i) P(ξ0=0)=1, (ii) (ξt)t[0,1] has independent increments, and (iii) for s<t, ξt-ξs is a normal random variable with mean 0 and variance t-s. (Namely, Brownian motion with starting point 0.) Because ξt-ξs has mean 0 and variance t-s, we calculate (cf. Isserlis’s theorem)


Thus using the Kolmogorov continuity theorem with α=4, β=1, K=3, there is a unique W𝒫E such that for all t1<<td,


i.e. for Bd,


For t1<<td and Bd, with T:dd defined by T(x1,,xd)=(x1,x2-x1,,xd-xd-1),


Hence, because ξt1,ξt2-ξt1,,ξtd-ξtd-1 are independent,


which means that the random variables πt1,πt2-πt1,,πtd-πtd-1 are independent.

If s<t and B1,B2, and for T:22 defined by T(x,y)=(x,y-x),

W((πs,πt-πs)(B1,B2)) =W(T(πs,πt)(B1,B2))

which implies that (πt-πs)*W=(ξt-ξs)*P, and because ξt-ξs is a normal random variable with mean 0 and variance t-s, so is πt-πs.



(E,E,W) is a probability space, and the stochastic process (πt)t[0,1] is a Brownian motion.

3 Interpolation and continuous stochastic processes

Let (ξt)t[0,1] be a continuous stochastic process with state space and sample space (Ω,,P). To say that the stochastic process is continuous means that for each ωΩ the map tξt(ω) is continuous [0,1]. Define ξ:ΩE by


For t[0,1] and B a Borel set in ,


and because ξt:(Ω,)(,) is measurable this belongs to . But by Theorem 4, E is generated by the collection {πt-1B:t[0,1],B}. Now, for f:XY and for a nonempty collection of subsets of Y,77 7 Charalambos D. Aliprantis and Kim C. Border, Infinite Dimensional Analysis: A Hitchhiker’s Guide, third ed., p. 140, Lemma 4.23.


Therefore ξ-1(E), which means that ξ:(Ω,)(E,E) is measurable. This means that a continuous stochastic proess with index set [0,1] induces a random variable with state space E. Then the pushforward measure of P by ξ is a Borel probability measure on E. We shall end up constructing a sequence of pushforward measures from a sequence of continuous stochastic processes, that converge in 𝒫E to Wiener measure W.

Let (Xn)n1 be a sequence of independent identically distributed random variables on a sample space (Ω,,P) with E(Xn)=0 and V(Xn)=1, and let S0=0 and


Then E(Sk)=0 and V(Sk)=k. For t0 let


Thus, for k0 and ktk+1,

Yt =Sk+(t-k)Xk+1

For each ωΩ, the map tYt(ω) is piecewise linear, equal to Sk(ω) when t=k, and in particular it is continuous. For n1, define

Xt(n)=n-1/2Ynt=n-1/2S[nt]+n-1/2(nt-[nt])X[nt]+1,t[0,1]. (1)

For 0kn,


For each n1, (Xt(n))t[0,1] is a continuous stochastic process on the sample space (Ω,,P), and we denote by Pn𝒫E the pushforward measure of P by X(n).

4 Donsker’s theorem

Lemma 6.

If Zn and Un are random variables with state space d such that ZnZ in distribution and Un0 in distribution, then Zn+Un0 in distribution.

If Zn are random variables with state space that converge in distribution to some random variable Z and cn are real numbers that converge to some real number c, then cnZncZ in distribution.

For σ0, let νσ2 be the Gaussian measure on with mean 0 and variance σ2. The characteristic function of νσ2 is, for σ>0,


and ν~0(ξ)=1. One checks that c*ν1=νc2 for c0.

In following theorem and in what follows, X(n) is the piecewise linear stochastic process defined in (1). We prove that a sequence of finite-dimensional distributions converge to a Gaussian measure.88 8 Bert Fristedt and Lawrence Gray, A Modern Approach to Probability Theory, p. 368, §19.1, Lemma 1.

Theorem 7.

For 0t0<t1<t1<<td1, the random vectors


converge in distribution to νt1-t0νtd-td-1 as n.


For 0<jd and n1 let


and for 0j<d and n1 let


with which

(Xt1(n)-Xt0(n),,Xtd(n)-Xtd-1(n)) =(Xr1,n(n)-Xs0,n(n),,Xrd,n(n)-Xsd-1,n(n))

Because E(Xt(n))=0,




and because 0ntj-[ntj]<1 this tends to 0 as n. Likewise, V(Vj,n)0 as n.

For 1jd,

Xrj,n(n)-Xsj-1,n(n) =n-1/2S[nrj,n]+n-1/2(nrj,n-[nrj,n])X[nrj,n]+1

By the central limit theorem,


in distribution as n. But


as n, and (tj-tj-1)*1/2ν1=νtj-tj-1, so by Lemma 6,


in distribution as n.

For sufficiently large n, depending on t0,,td,


Check that (U1,n,,Ud,n)0 in probability and that (V0,n,,Vd-1,n)0 in probability, and hence these random vectors converge to 0 in distribution as n. The random variables Xr1,n(n)-Xs0,n(n),,Xrd,n(n)-Xsd-1,n(n) are independent, and therefore their joint distribution is equal to the product of their distributions. Now, if μn=μn1μnd and μnjμj as n, 1jd, then for ξd,

μ~n(ξ) =μ~n1(ξ1)μ~nd(ξd)

as n, and therefore by Lévy’s continuity theorem, μnμ1μd as n. This means that the joint distribution of Xr1,n(n)-Xs0,n(n),,Xrd,n(n)-Xsd-1,n(n) converges to


as n. Because (U1,n,,Ud,n)0 in distribution as n and (V0,n,,Vd-1,n)0 in distribution as n, applying Lemma 6 we get that


in distribution as n, completing the proof. ∎

Let t0=0 and let 0<t1<<td1. As X0(n)=0, the above lemma tells us that


in distribution as n. Define g:dd by


The function g is continuous and satisfies


Then by the continuous mapping theorem,

(Xt1(n),Xt2(n),,Xtd(n))g*(νt1νt2-t1νtd-td-1) (2)

in distribution as n.99 9 Allan Gut, Probability: A Graduate Course, second ed., p. 245, Chapter 5, Theorem 10.4.

We prove a result that we use to prove the next lemma, and that lemma is used in the proof of Donsker’s theorem.1010 10 Ioannis Karatzas and Steven E. Shreve, Brownian Motion and Stochastic Calculus, second ed., p. 68, Lemma 4.18.

Lemma 8.

For ϵ>0,

limδ0lim supn1δP(max1j[nδ]+1|Sj|>ϵn1/2)=0.

For each δ>0, by the central limit theorem,


in distribution as n, where Z*P=ν1. Because ([nδ]+1)1/2(nδ)1/21 as n, by Lemma 6 we then get that


in distribution as n. Now let λ>0, and there is a sequence ϕk in Cb() such that ϕk1(-,-λ][λ,)=χλ pointwise as k. For each k, writing X=S[nδ]+1, using the change of variables formula,

P(|X|λ(nδ)1/2) =Ωχλ(nδ)1/2(X(ω))𝑑P(ω)

Therefore, by the continuous mapping theorem,

lim supnP(|S[nδ]+1|λ(nδ)1/2) limnE(ϕk((nδ)-1/2S[nδ]+1))

Because ϕkχλ pointwise as k, using the monotone convergence theorem and then using Chebyshev’s inequality,


We have established that for each λ>0,

lim supnP(|S[nδ]+1|λ(nδ)1/2)λ-3E|Z|3. (3)



For 0<δ<ϵ2/2, it is a fact that


If τ(ω)=j and |S[nδ]+1(ω)|<n1/2(ϵ-(2δ)1/2) then


But by Chebyshev’s inequality and the fact that the random variables X1,X2, are independent with mean 0 and variance 1,








Now using (3) with λ=(ϵ-(2δ)1/2)δ-1/2,

lim supnP(|S[nδ]+1|(ϵ-(2δ)1/2)δ-1/2(nδ)1/2)(ϵ-(2δ)1/2)-3δ3/2E|Z|3,


lim supnP(max0j[nδ]+1|Sj|>n1/2ϵ)2(ϵ-(2δ)1/2)-3δ3/2E|Z|3.

Dividing both sides by δ and then taking δ0 we obtain the claim. ∎

We prove one more result that we use to prove Donsker’s theorem.1111 11 Ioannis Karatzas and Steven E. Shreve, Brownian Motion and Stochastic Calculus, second ed., p. 69, Lemma 4.19.

Lemma 9.

For T>0 and ϵ>0,

limδ0lim supnP(max0k[nT]+1max1j[nδ]+1|Sj+k-Sk|>n1/2ϵ)=0.

For 0<δT, let m=T/δ, so T/m<δT/(m-1). Then


so for all nnδ it is the case that [nT]+1<([nδ]+1)m. Suppose that ωΩ is such that there are 1j[nδ]+1 and 0k[nT]+1 satisfying


and then let p=[k/([nδ]+1)], which satisfies 0pm-1 and


Because 1j[nδ]+1, either




We separate the first case into the cases




and we separate the second case into the cases






It follows that1212 12 This should be worked out more carefully. In Karatzas and Shreve, there is m+1 where I have m.


For 0pm-1,




Lemma 8 tells us

limδ0lim supn1δP(max1j[nδ]+1|Sj|>13n1/2ϵ)=0,

and because mTδ+1=T+δδ,

limδ0lim supnP{max1j[nδ]+1max0k[nT]+1|Sj+k-Sk|>n1/2ϵ}=0,

proving the claim. ∎

In the following, Pn𝒫E denotes the pushforward measure of P by X(n), for X(n) defined in (1). We now prove Donsker’s theorem.1313 13 Ioannis Karatzas and Steven E. Shreve, Brownian Motion and Stochastic Calculus, second ed., p. 70, Theorem 4.20.

Theorem 10 (Donsker’s theorem).



We shall use Theorem 3 to prove that Γ={Pn:n1} is relatively compact in 𝒫E. For n1,


thus the first condition of Theorem 3 is satisfied with Mϵ=0. For the second condition of Theorem 3 to be satisfied it suffices that for each ϵ>0,

limδ0lim supnP(sup0s,t1,|s-t|δ|X(n)(s)-X(n)(t)|>ϵ)=0.




sup0s,tn,|s-t|nδ|Ys-Yt| sup0s,tn,|s-t|nδ|Y-s-Yt|

so applying Lemma 9,

limδ0lim supnP(sup0s,t1,|s-t|δ|Xs(n)-Xt(n)|>ϵ)limδ0lim supnP(max1j[nδ]+1max0kn+1|Sj+k-Sk|>n1/2ϵ)0,

from which we get that Γ is tight in 𝒫E. ∎