Stochastic process

A stochastic process is a random function. In practical applications, the domain over which the function is defined is a time interval (a stochastic process of this kind is called a time series[?] in applications) or a region of space (a stochastic process being called a random field[?]). Familiar examples of time series include stock market and exchange rate fluctuations, signals such as speech, audio and video; medical data such as a patient's EKG, EEG, blood pressure or temperature; and random movement such as Brownian motion or random walks. Examples of random fields include static images, random topographies (landscapes), or composition variations of an inhomogeneous material.

Mathematically, if

f : D → R

is a random function with domain D and range R, the image of each point of D, f(x), is a random variable with values in R.

Of course, the mathematical definition of a function includes the case "a function from {1,...,n} to R is a vector in Rⁿ", so multivariate random variables are a special case of stochastic processes.

For our first infinite example, take the domain to be N, the natural numbers, and our range to be R, the real numbers. Then, a function f : N → R is a sequence of real numbers, and the following questions arise:

1. How is a random sequence specified?
2. How do we find the answers to typical questions about sequences, such as
   1. what is the probability distribution of the value of f(i)?
   2. what is the probability that f is bounded?
   3. what is the probability that f is monotonic?
   4. what is the probability that f(i) has a limit as i→∞?
   5. if we construct a series from f(i), what is the probability that the series converges[?]? What is the probability distribution of the sum?

Another important class of examples is when the domain is not a discrete space such as the natural numbers, but a continuous space[?] such as the unit interval [0,1], the positive real numbers [0,∞) or the entire real line, R. In this case, we have a different set of questions that we might want to answer:

1. How is a random function specified?
2. How do we find the answers to typical questions about functions, such as
   1. what is the probability distribution of the value of f(x) ?
   2. what is the probability that f is bounded/integrable[?]/continuous/differentiable...?
   3. what is the probability that f(x) has a limit as x→∞ ?
   4. what is the probability distribution of the integral <int>_a^bf(x)dx ?

Constructing stochastic processes: the Kolmogorov extension

In the ordinary axiomatization of probability theory by means of measure theory, the problem is to construct a sigma-algebra of measurable subsets[?] of the space of all functions, and then put a finite measure on it. For this purpose one traditionally uses a method called Kolmogorov extension.

The Kolmogorov extension proceeds along the following lines: assuming that a probability measure on the space of all functions f : X → Y exists, then it can be used to specify the probability distribution of finite-dimensional random variables [f(x₁),...,f(x_n)]. Now, from this n-dimensional probability distribution we can deduce an (n-1)-dimensional marginal probability distribution[?] for [f(x₁),...,f(x_n-1)]. There is an obvious compatibility condition, namely, that this marginal probability distribution be the same as the one derived from the full-blown stochastic process. When this condition is expressed in terms of probability densities, the result is called the Chapman-Kolmogorov equation[?].

The Kolmogorov extension theorem[?] guarantees the existence of a stochastic process with a given family of finite-dimensional probability distributions satisfying the Chapman-Kolmogorov compatibility condition.

Separability, or what the Kolmogorov extension does not provide

Recall that, in the Kolmogorov axiomatization, measurable[?] sets are the sets which have a probability or, in other words, the sets corresponding to yes/no questions that have a probabilistic answer.

The Kolmogorov extension starts by declaring to be measurable all sets of functions where finitely many coordinates [f(x₁),...,f(x_n)] are restricted to lie in measurable subsets of Yⁿ. In other words, if a yes/no question about f can be answered by looking at the values of at most finitely many coordinates, then it has a probabilistic answer.

In measure theory, if we have a countably infinite collection of measurable sets, then the union and intersection of all of them is a measurable set. For our purposes, this means that yes/no questions that depend on countably many coordinates have a probabilistic answer.

The good news is that the Kolmogorov extension makes it possible to construct stochastic processes with fairly arbitrary finite-dimensional distributions. Also, every question that one could ask about a sequence has a probabilistic answer when asked of a random sequence. The bad news is that certain questions about functions on a continuous domain don't have a probabilistic answer. One might hope that the questions that depend on uncountably many values of a function be of little interest, but the really bad news is that virtually all concepts of calculus are of this sort. For example:

1. boundedness
2. continuity
3. differentiability

all require knowledge of uncountably many values of the function.

One solution to this problem is to require that the stochastic process be separable. In other words, that there be some countable set of coordinates {f(x_i)} whose values determine the whole random function f.

Interesting special cases

homogeneous processes[?]

processes where the domain has some symmetry and the finite-dimensional probability distributions also have that symmetry. Special cases include stationary processes[?], also called time-homogeneous.

processes with independent increments[?]

processes where the domain is at least partially ordered and, if x₁ <...< x_n, all the variables f(x_k+1)-f(x_k) are independent. Markov chains are a special case.

point processes[?]

random arrangements of points in a space S. They can be modelled as stochastic processes where the domain is a sufficiently large family of subsets of S, ordered by inclusion; the range is the set of natural numbers; and, if A is a subset of B, f(A) ≤ f(B) with probability 1.

Gaussian processes[?]

processes where all linear combinations of coordinates are Gaussian random variables.

Poisson processes

Gauss-Markov processes

processes that are both Gaussian and Markov