from homework I got some weeks ago, we were asked if the bivariate vectors of $(X_t, X_s)$ and $(X_s, X_t)$ from strictly stationary Time Series$X_t$ do not have the same distribution. I thought they should, but the solution is that they do not.
Unfortunately, the solution only says False.
Can someone explain to me why?
The reason I thought it must be was almost trivial, since for strict stationarity I thought that $X_t, X_s \sim F$ for some distribution $F$.
Many thanks in advance!
Take a time series such that $X_t$ is uniformly distributed on $\{1,2,3\}$ and $X_{t+1}=\operatorname{mod}(X_t,3)+1$, i.e. you set $X_{t+1}=X_t+1$ if $X_t\in\{1,2\}$ and $X_{t+1}=1$ if $X_t=3$.
Then $(X_t,X_{t+1})$ is uniformly distributed on $\{(1,2), (2,3), (3,1)\}$ but $(X_{t+1},X_t)$ is uniformly distributed on $\{(1,3), (2,1), (3,2)\}$.