# Autocovariance

*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