# Autocovariance

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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=45247
This article was adapted from an original article by A.V. Prokhorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article