The Mathematics of ChanceCHANCE / MATHEMATICS

The Mathematics of Chance / ContentsPage progress

Chapter ownership and proof scope

Primary proof chapter: Chapter 23

Related tools and applications: Chapter 8

Tools needed by the individual arguments

Measure extension, Gaussian vectors, consistent finite-dimensional distributions, Kolmogorov's continuous modification theorem, and measurability of the path map.

These states describe the existing material; they do not certify a complete theorem treatment or final mathematical review.

Advanced Reading: Continuous Versions and Wiener Measure

4.8.1 Does Continuous Brownian Motion Really Exist?

The functional limit theorem for random walks has already suggested one route to a construction.

But there is another very important approach.

We can start with the covariance kernel

K(s,t)=\min(s,t).

For any finite collection of times t_1,\ldots,t_m, the matrix

\left(K(t_i,t_j)\right)_{i,j=1}^m

is positive semidefinite.

Indeed, using

K(s,t)=\int_0^\infty \mathbf1_{\{u\le s\}}\mathbf1_{\{u\le t\}}\,du,

we have

\begin{aligned} \sum_{i,j=1}^m a_ia_jK(t_i,t_j) &=\int_0^\infty \left(\sum_{i=1}^m a_i\mathbf1_{\{u\le t_i\}}\right)^2du\\ &\ge0. \end{aligned}

These matrices can therefore serve as covariance matrices of centered Gaussian vectors.

They also satisfy the consistency conditions required when time coordinates are deleted or reordered.

The Kolmogorov extension theorem thus allows us to construct a Gaussian process with these finite-dimensional distributions.

But one last step remains.

We must prove that it has a continuous modification.

Since

B_t-B_s\sim N(0,t-s),

we have

\mathbb E[|B_t-B_s|^4] = 3|t-s|^2.

This satisfies a moment condition of the kind required by Kolmogorov's continuity theorem.

We can therefore construct a version with continuous sample paths.

Only at this point have we truly obtained continuous-time Brownian motion. 〔41, 42〕

This proof route also illustrates the layered structure of modern probability theory:

\boxed{ \begin{gathered} \text{consistent finite-dimensional probability distributions}\\ \Downarrow\\ \text{existence of a stochastic process}\\ \Downarrow\\ \text{moment estimates for increments}\\ \Downarrow\\ \text{existence of a version with continuous paths} \end{gathered} }

These steps are distinct theorems, and none can simply be omitted.

4.8.2 Wiener Measure: An Entire Function Can Be a Random Variable

We now know that the sample paths of Brownian motion almost surely belong to

C([0,1]).

The most natural idea is therefore to take continuous functions themselves as the outcomes in a probability space.

Let

\Omega=C_0([0,1]) = \{f\in C([0,1]):f(0)=0\}.

and equip it with the Borel \sigma-algebra generated by the uniform norm.

We want to construct a probability measure \mathbb W on this function space such that the coordinate maps

B_t(\omega)=\omega(t)

form a standard Brownian motion.

This measure is called Wiener measure.

We can then assign probabilities to events in the entire function space.

For example:

A=\left\{ \omega:\sup_{0\le t\le1}\omega(t)\ge1 \right\}.

This is no longer an event concerning a single time.

It asks whether the whole path ever reaches a given height.

We can write

\mathbb W(A) = P\left(\sup_{0\le t\le1}B_t\ge1\right).

Notice that this answers the question left open at the end of Chapter 1.

What should we do when outcomes can no longer be described by counting?

Our sample space is now not even an interval, but an infinite-dimensional function space.

Probability can still be defined because its true foundation is a measure satisfying appropriate axioms, rather than “the number of favorable outcomes divided by the total number of outcomes.”

Here a random variable can be a real number, a continuous function, or even a more complicated random geometric object.

Modern probability theory begins to free itself from dependence on the particular form of its random outcomes.

This is an indispensable foundation for our later study of random trees and the CRT.