Autocovariance Matrix for Stationary Process is Variance by Autocorrelation Matrix

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

Let $\sequence {s_n}$ be a sequence of $n$ successive values of $T$:

$\sequence {s_n} = \tuple {z_1, z_2, \dotsb, z_n}$

Let $\boldsymbol \Gamma_n$ denote the autocovariance matrix associated with $S$ for $\sequence {s_n}$.

Let $\mathbf P_n$ denote the autocorrelation matrix associated with $S$ for $\sequence {s_n}$.


$\boldsymbol \Gamma_n = \sigma_z^2 \mathbf P_n$

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


From Autocorrelation of Strictly Stationary Stochastic Process:

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

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

$\gamma_0 = \sigma_z^2$


$\gamma_k = \sigma_z^2 \rho_k$

and the result follows.



Part $\text {I}$: Stochastic Models and their Forecasting:
$2$: Autocorrelation Function and Spectrum of Stationary Processes:
$2.1$ Autocorrelation Properties of Stationary Models:
$2.1.3$ Positive Definiteness and the Autocovariance Matrix