I have a pretty basic question: If a Gaussian random process is uncorrelated with itself, does it imply that it is stationary? I think not, but I wanted to confirm my assertion.
Edit: Uncorrelated with itself means that any two samples of the random process (taken at different times) have zero covariance. In other words:
$\text{Cov}_X(t_1, t_2)=\text{Cov}(X(t_1), X(t_2))=0~\text{for all}~t_1 \neq t_2$