Identically distributed variables enough to establish stationarity? (time series)

Question

If you can prove a sequence (X_t) of random variables to be identically distributed for all t, will this be enough to establish stationarity? Or put differently, can you give example of a non-stationary series with identically distributed marginals for all t? Thanks,

Answer 1

The answer of user940 in this thread gives an example of vector $\left(X,Y\right)$ such that $X$ and $Y$ have the same distribution but the vectors $\left(X,Y\right)$ and $\left(Y,X\right)$ do not.

Then define $X_1=X$, $X_2=Y$, $X_3=X$ and for the other $t$, we can choose $X_t=X$ for example. Then the vectors $\left(X_1,X_2\right)$ and $\left(X_2,X_3\right)$ do not have the same distribution but the sequence $\left(X_t\right)_{t\geqslant 1}$ is identically distributed.