View source for Stochastic process, generalized

A [[Stochastic process|stochastic process]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090220/s0902201.png" /> depending on a continuous (time) argument <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090220/s0902202.png" /> and such that its values at fixed moments of time do not, in general, exist, but the process has only  "smoothed values"  <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090220/s0902203.png" /> describing the results of measuring its values by means of all possible linear measuring devices with sufficiently smooth weight function (or impulse transition function) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090220/s0902204.png" />. A generalized stochastic process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090220/s0902205.png" /> is a continuous linear mapping of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090220/s0902206.png" /> of infinitely-differentiable functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090220/s0902207.png" /> of compact support (or any other space of test functions used in the theory of generalized functions) into the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090220/s0902208.png" /> of random variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090220/s0902209.png" /> defined on some probability space. Its realizations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090220/s09022010.png" /> are ordinary generalized functions of the argument <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090220/s09022011.png" />. Ordinary stochastic processes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090220/s09022012.png" /> can also be regarded as generalized stochastic processes, for which

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this is particularly useful in combination with the fact that a generalized stochastic process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090220/s09022014.png" /> always has derivatives <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090220/s09022015.png" /> of any order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090220/s09022016.png" />, given by

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(see, for example, [[Stochastic process with stationary increments|Stochastic process with stationary increments]]). The most important example of a generalized stochastic process of non-classical type is that of [[White noise|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|Random field, generalized]].



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====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  I.M. Gel'fand,   N.Ya. Vilenkin,   "Generalized functions. Applications of harmonic analysis" , '''4''' , Acad. Press  (1964)  (Translated from Russian)</TD></TR></table>