Sufficient Conditions for Weak Stationarity of Order 2

Theorem

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

Let the mean of $S$ be fixed.

Let the autocovariance matrix of $S$ be of the form:

$\boldsymbol \Gamma_n = \begin {pmatrix}

\gamma_0 & \gamma_1 & \gamma_2 & \cdots & \gamma_{n - 1} \\ \gamma_1 & \gamma_0 & \gamma_1 & \cdots & \gamma_{n - 2} \\ \gamma_2 & \gamma_1 & \gamma_0 & \cdots & \gamma_{n - 3} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ \gamma_{n - 1} & \gamma_{n - 2} & \gamma_{n - 3} & \cdots & \gamma_0 \end {pmatrix} = \sigma_z^2 \mathbf P_n = \begin {pmatrix} 1 & \rho_1 & \rho_2 & \cdots & \rho_{n - 1} \\ \rho_1 & 1 & \rho_1 & \cdots & \rho_{n - 2} \\ \rho_2 & \rho_1 & 1 & \cdots & \rho_{n - 3} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ \rho_{n - 1} & \rho_{n - 2} & \rho_{n - 3} & \cdots & 1 \end {pmatrix}$

Then $S$ is weakly stationary of order $2$.

Proof

Follows from the definition of weak stationarity.

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: Weak stationarity