Random function

A function of an arbitrary argument (defined on the set of its values, and taking numerical values or, more generally, values in a vector space) whose values are defined in terms of a certain experiment and may vary with the outcome of this experiment according to a given probability distribution. In probability theory, attention centres on numerical (that is, scalar) random functions ; a random vector function can be regarded as the aggregate of the scalar functions , where ranges over the finite or countable set of components of , that is, as a numerical random function on the set of pairs , , .

When is finite, is a finite set of random variables, and can be regarded as a multi-dimensional (vector) random variable characterized by a multi-dimensional distribution function. When is infinite, the case mostly studied is that in which takes numerical (real) values; in this case, usually denotes time, and is called a stochastic process, or, if takes only integral values, a random sequence (or time series). If the values of are the points of a manifold (such as a -dimensional Euclidean space ), then is called a random field.

The probability distribution of the values of a random function defined on an infinite set is characterized by the aggregate of finite-dimensional probability distributions of sets of random variables corresponding to all finite subsets of , that is, the aggregate of corresponding finite-dimensional distribution functions , satisfying the consistency conditions:

(1)

(2)

where is an arbitrary permutation of the subscripts . This characterization of the probability distribution of is sufficient in all cases when one is only interested in events depending on the values of on countable subsets of . But it does not enable one to determine the probability of properties of that depend on its values on a continuous subset of , such as the probability of continuity or differentiability, or the probability that on a continuous subset of (see Separable process).

Random functions can be described more generally in terms of aggregates of random variables defined on a fixed probability space (where is a set of points , is a -algebra of subsets of and is a given probability measure on ), one for each point of . In this approach, a random function on is regarded as a function of two variables and which is -measurable for every (that is, for fixed it reduces to a random variable defined on the probability space ). By taking a fixed value of , one obtains a numerical function on , called a realization (or sample function or, when denotes time, a trajectory) of ; and induce a -algebra of subsets and a probability measure defined on it in the function space of realizations , whose specification can also be regarded as equivalent to that of the random function. The specification of a random function as a probability measure on a -algebra of subsets of the function space of all possible realizations can be regarded as a special case of its general specification as a function of two variables (where belongs to the probability space in which ), that is, elementary events (points in the given probability space) are identified at the outset with the realizations of . On the other hand, it is also possible to show that any other way of specifying can be reduced to this form using a special determination of a probability measure on . In particular, Kolmogorov's fundamental theorem on consistent distributions (see Probability space) shows that the specification of the aggregate of all possible finite-dimensional distribution functions satisfying the above consistency conditions (1) and (2) defines a probability measure on the -algebra of subsets of the function space generated by the aggregate of cylindrical sets (cf. Cylinder set) of the form , where is an arbitrary positive integer and is an arbitrary Borel set of the -dimensional space of vectors .

For references see Stochastic process.

References

[a1]	J.L. Doob, "Stochastic processes" , Wiley (1953)
[a2]	M. Loève, "Probability theory" , Springer (1977)
[a3]	I.I. [I.I. Gikhman] Gihman, A.V. [A.V. Skorokhod] Skorohod, "The theory of stochastic processes" , 1 , Springer (1974) (Translated from Russian)
[a4]	A. Blanc-Lapierre, R. Fortet, "Theory of random functions" , 1–2 , Gordon & Breach (1965) (Translated from French)