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
- 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: $(2.1.6)$
- $2.1$ Autocorrelation Properties of Stationary Models:
- $2$: Autocorrelation Function and Spectrum of Stationary Processes:
- Part $\text {I}$: Stochastic Models and their Forecasting: