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Stochastic process, generalized

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A stochastic process depending on a continuous (time) argument and such that its values at fixed moments of time do not, in general, exist, but the process has only "smoothed values" describing the results of measuring its values by means of all possible linear measuring devices with sufficiently smooth weight function (or impulse transition function) . A generalized stochastic process is a continuous linear mapping of the space of infinitely-differentiable functions of compact support (or any other space of test functions used in the theory of generalized functions) into the space of random variables defined on some probability space. Its realizations are ordinary generalized functions of the argument . Ordinary stochastic processes can also be regarded as generalized stochastic processes, for which

this is particularly useful in combination with the fact that a generalized stochastic process always has derivatives of any order , given by

(see, for example, Stochastic process with stationary increments). The most important example of a generalized stochastic process of non-classical type is that of white noise. A generalization of the concept of a generalized stochastic process is that of a generalized random field.

For references, see Random field, generalized.


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References

[a1] I.M. Gel'fand, N.Ya. Vilenkin, "Generalized functions. Applications of harmonic analysis" , 4 , Acad. Press (1964) (Translated from Russian)
How to Cite This Entry:
Stochastic process, generalized. A.M. Yaglom (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Stochastic_process,_generalized&oldid=13436
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098