"strong stationary process" equivalent to a process with "indentically distributed" random variables?

Question

I want to know if a strong stationary process is equivalent to a process with identically distributed RVs.

In another words : if A is the set of s.s.s processes and B is the set of processes with identically distributed RVs, is it true that A = B ?

Answer 1

No, they are not the same. In a strong stationary process $(X_1,X_2,\dots)$ it is the case that all of the $X_n$'s have the same distribution. However, the converse does not hold.

To see why, take $X_2,X_3,\dots$ to be IID (whatever distribution you like, but not constant), and $X_1=X_2$. Then all of these terms are identically distributed, but this is not a strong stationary sequences because, for instance, $(X_1,X_2)$ is not equal in distribution to $(X_2,X_3)$.