I'm having trouble grasping the autocorrelation function for stationary signals, both strict stationary and WSS.
First for strict sense, we have $$\forall(\tau,t_1, \ldots, t_n) \in \mathbb{R} \land \forall n \in \mathbb{N} \ \ \ F_{X}(X_{t_1+\tau} ,\ldots, X_{t_n+\tau}) = F_{X}(X_{t_1},\ldots, X_{t_n}) \ \ $$ For unconditional cumulative distribution function $F_X$
This implies that all time instants are identically distributed $$\forall(t_1,t_2) \ \ F_X(X_{t_1})=F_X(X_{t_2})$$
My question then is, how is it possible for the autocorrelation to be anything other than a constant? The only thing I can think of is each time instant random variable $X_t$ is not independent, so the product distribution function would be different for different time instances and $$E[X_{t_1}X_{t_2}]\ne E[X_{t_1}X_{t_3}]$$ if they were independent then the product distribution would be the same for all time instants and therefore the expected value as well giving a constant autocorrelation.
Additionally, for WSS, I can't conceptualize how autocorrelation could be independent of time, but still possibly dependent on shift. In this case however, each time instant is not necessarily identically distributed, only they have the same mean, but then it would seem the autocorrelation would depend on time.
Altogether, it seems that if the autocorrelation is time independent, then each time instant random variable is identically distributed (is this not the case?) and if they are identically distributed, the only way shift can affect the value is if
each time instant random variable has a different dependence on the other time instant random variables, but the dependence is consistent for all the random variables with respect to their "relative neighbors"
Is this understanding ok, or am I completely off the mark? Can someone provide an example where the autocorrelation of the stationary process is time independent, but does vary with shift?