Mathematically, if
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 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:
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:
In the ordinary axiomatization of probability theory by means of measure theory, the problem is to construct a sigmaalgebra 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 finitedimensional random variables [f(x_{1}),...,f(x_{n})]. Now, from this ndimensional probability distribution we can deduce an (n1)dimensional marginal probability distribution[?] for [f(x_{1}),...,f(x_{n1})]. There is an obvious compatibility condition, namely, that this marginal probability distribution be the same as the one derived from the fullblown stochastic process. When this condition is expressed in terms of probability densities, the result is called the ChapmanKolmogorov equation[?].
The Kolmogorov extension theorem[?] guarantees the existence of a stochastic process with a given family of finitedimensional probability distributions satisfying the ChapmanKolmogorov compatibility condition.
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(x_{1}),...,f(x_{n})] are restricted to lie in measurable subsets of Y^{n}. 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 finitedimensional 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:
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.
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