# Difference between revisions of "Autocovariance"

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+ | $#C+1 = 8 : ~/encyclopedia/old_files/data/A013/A.0103990 Autocovariance | ||

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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 | + | The [[Covariance|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|Stationary stochastic process]]). For such processes the autocovariance depends only on $ h $ | ||

+ | and differs from the [[Auto-correlation|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" . |

## 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