Random Function


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random function

[′ran·dəm ′fəŋk·shən]
(mathematics)
A function whose domain is an interval of the extended real numbers and has range in the set of random variables on some probability space; more precisely, a mapping of the cartesian product of an interval in the extended reals with a probability space to the extended reals so that each section is a random variable.

Random Function

 

If a function of an arbitrary argument t is defined on the set T of the values of t and assumes numerical values or, more generally, values from some vector space, the function is said to be a random function if its values are determined by some trial and can differ depending on the outcome of the trial. It is also required that there exist a definite probability distribution function for the values. If the set T is finite, the random function is a finite set of random variables; this set can be regarded as a single random vector quantity.

The most thoroughly studied random function with an infinite T is the important special case where t assumes numerical values and is time. The random function X(t) in this case is called a stochastic process; when t assumes only integral values, the terms “random sequence” and “time series” are also sometimes applied to X(t). If the values of t are points in some region of a multidimensional space, the random function is called a random field. Typical examples of random functions that are not stochastic processes are the velocity, pressure, and temperature fields of a turbulent flow of a liquid or gas and the height z of the agitated surface of the sea or the surface of an artificial rough plate.

The mathematical theory of random functions coincides with the theory of probability distribution functions in the function space of the values of X(t). These distribution functions can be specified by the set of finite-dimensional probability distribution functions for the sets of random variables X(t1), X(t2), …, X(tn) corresponding to all possible finite subsets (t1, t2,…, tn) of points of T. Alternatively, the distribution functions can be specified by the characteristic functional of the random function X(t); this characteristic functional is the mathematical expectation of the random variable il[X(t)], where l[X(t)] is a linear functional of X(t) of general form. Much progress has been made in the theory of homogeneous random fields, which are a special class of random functions; this class is a generalization of the class of stationary stochastic processes.

REFERENCES

Vybrosy sluchainykh polei: Sb. st. Moscow, 1972.
Yaglom (Iaglom), A. M. “Second-order Homogeneous Random Fields.” In Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, vol. 2. Berkeley-Los Angeles, 1961.
Whittle, P. “Stochastic Processes in Several Dimensions.” Bulletin of the Institute of Statistics, 1963, vol. 40.

A. M. IAGLOM

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