I have a question about stochastic processes in general. This question is out of pure curiosity. I am self-teaching myself this stuff as a hobby so this question is not related to any particular problem.
Let's say that there is some Gaussian wide seance stationary process $X(t)$. Are realizations at different times independent? In other words, is $X(2)$ necessarily independent of $X(1)$? Is $X$ independent identically distributed at various times?
I appreciate any guidance. The book I have (Essentials of Stochastic Processes) is not clear on this point and I am unable to find much info on various internet searches.