Autocovariance Matrix for Stationary Process is Variance by Autocorrelation Matrix

Theorem

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 $\sequence {s_n}$.

Then:

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

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

Proof

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:

$\gamma_0 = \sigma_z^2$

Hence:

$\gamma_k = \sigma_z^2 \rho_k$

and the result follows.

$\blacksquare$

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:

$2.1.3$ Positive Definiteness and the Autocovariance Matrix