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 of order $2$.

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.