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Difference between revisions of "Autocovariance"

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''of a stochastic process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013990/a0139901.png" />''
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The [[Covariance|covariance]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013990/a0139902.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013990/a0139903.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013990/a0139904.png" /> denotes the mathematical expectation of a random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013990/a0139905.png" />, then the autocovariance equals
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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/a/a013/a013990/a0139906.png" /></td> </tr></table>
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''of a stochastic process  $  X _ {t} $''
  
The term  "autocovariance"  is usually applied to stationary (in the wide sense) stochastic processes (cf. [[Stationary stochastic process|Stationary stochastic process]]). For such processes the autocovariance depends only on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013990/a0139907.png" /> and differs from the [[Auto-correlation|auto-correlation]] by the presence of a single factor which is equal to the variance of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013990/a0139908.png" />. The terms  "covariance function"  and  "autocovariance function"  are used together with the term  "autocovariance" .
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The [[Covariance|covariance]] of  $  X _ {t} $
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and  $  X _ {t+h} $.
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If  $  {\mathsf E} X $
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denotes the mathematical expectation of a random variable  $  X $,
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then the autocovariance equals
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$$
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{\mathsf E} ( X _ {t} - {\mathsf E} X _ {t} ) ( X _ {t+h} - {\mathsf E}
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X _ {t+h} ).
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$$
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The term  "autocovariance"  is usually applied to stationary (in the wide sense) stochastic processes (cf. [[Stationary stochastic process|Stationary stochastic process]]). For such processes the autocovariance depends only on $  h $
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and differs from the [[Auto-correlation|auto-correlation]] by the presence of a single factor which is equal to the variance of $  X _ {t} $.  
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The terms  "covariance function"  and  "autocovariance function"  are used together with the term  "autocovariance" .

Latest revision as of 18:48, 5 April 2020


of a stochastic process $ X _ {t} $

The covariance of $ X _ {t} $ and $ X _ {t+h} $. If $ {\mathsf E} X $ denotes the mathematical expectation of a random variable $ X $, then the autocovariance equals

$$ {\mathsf E} ( X _ {t} - {\mathsf E} X _ {t} ) ( X _ {t+h} - {\mathsf E} X _ {t+h} ). $$

The term "autocovariance" is usually applied to stationary (in the wide sense) stochastic processes (cf. Stationary stochastic process). For such processes the autocovariance depends only on $ h $ and differs from the auto-correlation by the presence of a single factor which is equal to the variance of $ X _ {t} $. The terms "covariance function" and "autocovariance function" are used together with the term "autocovariance" .

How to Cite This Entry:
Autocovariance. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Autocovariance&oldid=13706
This article was adapted from an original article by A.V. Prokhorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article