Autocorrelation of Strictly Stationary Stochastic Process

Example of Strictly Stationary Stochastic Process
Let $S$ be a strictly stationary stochastic process giving rise to a time series $T$.

It is necessary that:
 * The autocorrelation between every two observations $z_t, z_{t + k}$ separated by a given lag $k$

is the same as:
 * the autocorrelation between every other two observations $z_{t + m}, z_{t + m + k}$separated by a given lag $k$

For such a strictly stationary stochastic process:
 * $\rho_k = \dfrac {\gamma_k} {\gamma_0}$

where $\gamma_k$ denotes the autocovariance of $S$.

Proof
The autocorrelation is defined as:


 * $\rho_k := \dfrac {\expect {\paren {z_t - \mu} \paren {z_{t + k} - \mu} } } {\sqrt {\expect {\paren {z_t - \mu}^2} \expect {\paren {z_{t + k} - \mu}^2} } }$

The autocovariance is defined as:


 * $\gamma_k := \expect {\paren {z_t - \mu} \paren {z_{t - k} - \mu} }$

Hence:


 * $\rho_k := \dfrac {\gamma_k} {\sqrt {\expect {\paren {z_t - \mu}^2} \expect {\paren {z_{t + k} - \mu}^2} } }$

Then we have that for a strictly stationary stochastic process:
 * $\expect {\paren {z_t - \mu}^2} = \sigma_t^2$

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

Thus:
 * $\rho_k := \dfrac {\gamma_k} {\sigma_t^2}$

Then we have that:


 * $\gamma_0 = \expect {\paren {z_t - \mu} \paren {z_{t + 0} - \mu} } = \expect {\paren {z_t - \mu}^2}$

from which it follows that:
 * $\sigma_t^2 = \gamma_0$

Hence:
 * $\rho_k = \dfrac {\gamma_k} {\gamma_0}$