Autocorrelation of Strictly Stationary Stochastic Process

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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:

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

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

Thus:

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

Then from Autocovariance at Zero Lag for Strictly Stationary Stochastic Process is Variance:

$\sigma_z^2 = \gamma_0$

Hence:

$\rho_k = \dfrac {\gamma_k} {\gamma_0}$

$\blacksquare$


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: $(2.1.6)$