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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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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090220/s09022013.png" /></td> </tr></table>
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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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A [[Stochastic process|stochastic process]]  $  X $
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depending on a continuous (time) argument  $  t $
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and such that its values at fixed moments of time do not, in general, exist, but the process has only  "smoothed values"   $  X ( \phi ) $
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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) $.  
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A generalized stochastic process  $  x ( \phi ) $
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is a continuous linear mapping of the space  $  D $
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of infinitely-differentiable functions  $  \phi $
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of compact support (or any other space of test functions used in the theory of generalized functions) into the space  $  L _ {0} $
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of random variables  $  X $
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defined on some probability space. Its realizations  $  x ( \phi ) $
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are ordinary generalized functions of the argument  $  t $.  
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Ordinary stochastic processes  $  X ( t) $
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can also be regarded as generalized stochastic processes, for which
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090220/s09022017.png" /></td> </tr></table>
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$$
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X ( \phi )  = \int\limits _ {- \infty } ^  \infty 
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\phi ( t) X ( t)  d t ;
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$$
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this is particularly useful in combination with the fact that a generalized stochastic process  $  X $
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always has derivatives  $  X  ^ {(} n) $
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of any order  $  n $,
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given by
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$$
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X  ^ {(} n) ( \phi )  = ( - 1 )  ^ {n} X ( \phi  ^ {(} n) )
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$$
  
 
(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.
 
(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]].
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====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>

Latest 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
This article was adapted from an original article by A.M. Yaglom (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article