Find an example of a stochastic process

Question

Give an example of a stochastic process $\{X_n : n \in \mathbb{Z} \}$ on $(\Omega,F, P)$ such that $P_{X_n}=P_{X_0}$ for all n, but $\{X_n \}$ is not stationary. Here $P_{X_n}$ denotes the push forward measure of $P$ under $X_n$.

I'm not sure how to start this. Any hints or examples will be appreciated! Thank you!

Answer 1

Let $X$ and $Y$ independent random variables with standard normal distribution. Then $X,Y,X,X,...$ is not stationary but it is identically distributed.