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: