Characterization of Stationary Gaussian Process

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

Let the the mean of $S$ be fixed.

Let the autocovariance matrix of $S$ also be fixed.

Then $S$ is stationary.

Proof
From Characterization of Multivariate Gaussian Distribution, the Gaussian distribution is completely characterized by its expectation and its variance.

The result follows.