Autocovariance at Zero Lag for Strictly Stationary Stochastic Process is Variance

Example of Strictly Stationary Stochastic Process

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

Then the autocovariance at zero lag is given by:

$\gamma_0 = \sigma_z^2$

where $\sigma_z^2$ is the variance of $S$.

Proof

By definition, 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.

For a strictly stationary stochastic process:

$\expect {\paren {z_t - \mu}^2} = \sigma_z^2$

where:

$\mu$ is the constant mean level of $S$

$\expect {\paren {z_t - \mu}^2}$ is the expectation of $\paren {z_t - \mu}^2$

$\sigma_z^2$ is the variance of $S$ and, for a strictly stationary stochastic process, is constant.

Hence we have that:

\(\ds \gamma_0\)

\(=\)

\(\ds \expect {\paren {z_t - \mu} \paren {z_{t + 0} - \mu} }\)

\(\ds \)

\(=\)

\(\ds \expect {\paren {z_t - \mu}^2}\)

\(\ds \)

\(=\)

\(\ds \sigma_z^2\)

$\blacksquare$

Sources

• 1994: George E.P. Box, Gwilym M. Jenkins and Gregory C. Reinsel: Time Series Analysis: Forecasting and Control (3rd ed.) ... (previous) ... (next):

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