Every reference I see about WSS stochastic processes states it as a definition, maybe solves a few examples, and moves on without saying much about it conceptually. Let's take the discrete case $\left\{\mathbf{X}_n : n \in \mathbb{Z}_0^\infty\right\}$.
Other than the fact that we can examine the power spectrum/cross spectrum and that the cross-correlations are sort of "time-invariant" and it's a function of a single variable, what more intuitive/conceptual things can you say about a WSS process?
Is it fair to say if a $\mathbf{X}$ is WSS, we can argue modeling the data with a Gaussian process is a decent model? What kind of things can we say about modeling just based on WSS?
EDIT: I have a hunch that modeling thing is a decent assumption, because Dube theorem 3.1 seems to be saying that any mean and covariance pair can be mapped to a Gaussian process (assuming the mean/covariance functions have certain properties, which I think just guarantee they can be generated by at least ONE stochastic process).