# Definition:Autocovariance

## Definition

Let $S$ be a stochastic process giving rise to a time series $T$.

The autocovariance of $S$ at lag $k$ is defined as:

$\gamma_k := \cov {z_t, z_{t + k} } = \expect {\paren {z_t - \mu} \paren {z_{t - k} - \mu} }$

where:

$z_t$ is the observation at time $t$
$\mu$ is the mean of $S$
$\expect \cdot$ is the expectation.

## Also see

• Results about autocovariance can be found here.

## Sources

Part $\text {I}$: Stochastic Models and their Forecasting:
$2$: Autocorrelation Function and Spectrum of Stationary Processes:
$2.1$ Autocorrelation Properties of Stationary Models:
$2.1.2$ Stationary Stochastic Processes: Autocovariance and autocorrelation coefficients: $(2.1.5)$