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A stochastic process is a random function. This means that, 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 R^n", so multidimensional 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 is f monotonic?
   4. what is the probability that f(i) has a limit as i->infty?
   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 , the positive real numbers [0,infty) 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->infty?

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 functions f : X -> Y exists, then it can be used to specify the probability distribution of finite-dimensional marginal random variables (f(x_1),...,f(x_n)). These finite-dimensional distributions must satisfy the Chapman-Kolmogorov compatibility condition.