# Difference between revisions of "Stochastic process, generalized"

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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. | (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]]. | For references, see [[Random field, generalized|Random field, generalized]]. | ||

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====Comments==== | ====Comments==== | ||

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====References==== | ====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> | <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> |

## Revision as of 08:23, 6 June 2020

A 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). 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.

#### Comments

#### 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.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Stochastic_process,_generalized&oldid=13436