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A [[Stochastic process|stochastic process]]  $  X $
depending on a continuous (time) argument  $  t $
and such that its values at fixed moments of time do not, in general, exist, but the process has only  "smoothed values"   $  X ( \phi ) $
describing the results of measuring its values by means of all possible linear measuring devices with sufficiently smooth weight function (or impulse transition function)  $  \phi ( t) $. 
A generalized stochastic process  $  x ( \phi ) $
is a continuous linear mapping of the space  $  D $
of infinitely-differentiable functions  $  \phi $
of compact support (or any other space of test functions used in the theory of generalized functions) into the space  $  L _ {0} $
of random variables  $  X $
defined on some probability space. Its realizations  $  x ( \phi ) $
are ordinary generalized functions of the argument  $  t $. 
Ordinary stochastic processes  $  X ( t) $
can also be regarded as generalized stochastic processes, for which

$$ 
X ( \phi )  =  \int\limits _ {- \infty } ^  \infty  
\phi ( t) X ( t)  d t ;
$$

this is particularly useful in combination with the fact that a generalized stochastic process  $  X $
always has derivatives  $  X  ^ {(n)}  $
of any order  $  n $, 
given by

$$ 
X  ^ {(n)} ( \phi )  =  ( - 1 )  ^ {n} X ( \phi  ^ {(n)}  )
$$

(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]].

====Comments====

====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>