Second Order Weakly Stationary Gaussian Stochastic Process is Strictly Stationary

Theorem

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

Let $S$ be weakly stationary of order $2$.

Then $S$ is strictly stationary.

Proof

By definition of a Gaussian process, the probability distribution of $T$ be a multivariate Gaussian distribution.

By definition, a Gaussian distribution is characterized completely by its expectation and its variance.

That is, its $1$st and $2$nd moments.

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: Weak stationarity