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Consider a strictly stationary process $X_t$, $t\in\mathbb{Z}_{\geq 1}$. Could you help me to disprove the following statement:

"For $t, s > 0$, the bivariate vectors $(X_s, X_t)$ and $(X_t, X_s)$ have the same distribution."

I think the statement is false in general but true for Gaussian processes. Can we find a counter-example which proves that the statement is false?

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    $\begingroup$ take a stationary but non-reversible markov chain, or look at rotations of circle, $X_{n+1} = e^{\theta}X_n$ where $X_i$ are uniform on circle $\endgroup$
    – mike
    Commented Jan 13, 2023 at 10:02
  • $\begingroup$ Thanks @mike for the helpful answer. Is it correct to assume that $\theta$ is a fixed parameter on $(0,2\pi)$? Maybe you could post your comment as a formal answer so I can mark your answer as "accepted answer". $\endgroup$
    – iom10
    Commented Jan 13, 2023 at 10:13

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