The Daniell-Kolmogorov extension theorem is one of the first deep theorems of the theory of stochastic processes. It provides existence results for nice probability measures on path (function) spaces. It is however non-constructive and relies on the axiom of choice. In what follows, in order to avoid heavy notations we restrict to the one dimensional case . The multidimensional extension is straightforward and let to the reader.
Definition. Let be a stochastic process. For we denote by the probability distribution of the random variable It is therefore a probability measure on . This probability measure is called a finite dimensional distribution of the process .
If two processes have the same finite dimensional distributions, then it is clear that the two processes induce the same distribution on the path space because cylinders generate the -algebra (see Lecture 2).
The finite dimensional distributions of a given process satisfy the two following properties: If and if is a permutation of the set , then:
Theorem (Daniell-Kolmogorov theorem). Assume that we are given for every a probability measure on . Let us assume that these probability measures satisfy:
Then, there is a unique probability measure on such that for , :
The Daniell-Kolmogorov theorem is often used to construct processes thanks to the following corollary:
Corollary. Assume given for every a probability measure on . Let us further assume that these measures satisfy the assumptions of the Daniell-Kolmogorov theorem. Then, there exists a probability space as well as a process defined on this space such that the finite dimensional distributions of are given by the ‘s.
Proof of the corollary:
As a probability space we chose
where is the probability measure given by the Daniell-Kolmogorov theorem. The canonical coordinate process defined on by satisfies the required property.
We now turn to the proof of the Daniell-Kolmogorov theorem. This proof proceeds in several steps.
Theorem (Caratheodory theorem). Let be a non-empty set and let be a family of subsets that satisfy:
- If , ;
- If , .
Let be the -algebra generated by . If is -additive measure on which is -finite, then there exists a unique -additive measure on such that for ,
As a first step, we prove the following fact:
Lemma. Let , be a sequence of Borel sets that satisfy Let us assume that for every a probability measure is given on and that these probability measures are compatible in the sense that
where . There exists a sequence of compact sets , , such that:
Proof of the lemma:
For every , we can find a compact set such that
It is easily checked that:
With this in hands, we can now turn to the proof of the Daniell-Kolmogorov theorem.
Proof of the Daniell-Kolmogorov theorem:
For the cylinder
where and where is a Borel subset of , we define
Thanks to the assumptions on the ‘s, it is seen that such a is well defined and satisfies:
The set of all the possible cylinders satisfies the assumption of Caratheodory’s theorem. Therefore, in order to conclude, we have to show that is -additive, that is, if is a sequence of pairwise disjoint cylinders and if is a cylinder then
This is the difficult part of the theorem. Since for ,
we just have to show that
The sequence is positive decreasing and therefore converges. Let assume that it converges toward . We shall prove that in that case
which is clearly absurd.
Since is a cylinder, the event only involves a coutable sequence of times and we may assume (otherwise we can add convenient other sets in the sequence of the 's) that every can be described as follows
where , , is a sequence of Borel sets such that
Since we assumed , we can use the previous lemma to construct a sequence of compact sets , , such that:
Since is non-empty, we pick
The sequence has a convergent subsequence that converges toward . The sequence has a convergent subsequence that converges toward . By pursuing this process, we obtain a sequence such that for every ,
is in , this leads to the expected contradiction. Therefore, the sequence converges toward , which implies the -additivity of
As it has been stressed, the Daniell-Kolmogorov theorem is the basic tool to prove the existence of a stochastic process with given finite dimensional distributions. As an example, let us illustrate how it may be used to prove the existence of the so-called Gaussian processes.
Definition. A real-valued stochastic process defined on is said to be a Gaussian process if all the finite dimensional distributions of are Gaussian random variables.
If is a Gaussian process, its finite dimensional distributions can be characterized, through Fourier transform, by its mean function
and its covariance function
We can observe that the covariance function is symmetric and positive, that is for and ,
Conversely, as an application of the Daniell-Kolmogorov theorem, we let the reader prove as an exercise the following proposition.
Proposition. Let and let be a symmetric and positive function. There exists a probability space and a Gaussian process defined on it, whose mean function is and whose covariance function is .