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Let $\{X_t\}_{t=1}^{\infty}$ be a Markov Chain.

An initial marginal distribution $\pi^T$ for a markov chain is a stationary distribution if $\pi^TP = \pi^T$.

My understanding of this is that if the initial marginal distribution is stationary, then the marginal distributions of the chain is the same at all time points after the starting time. Does this mean that $X_1,X_2,X_3 \ldots$ are all iid Random Variables?

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    $\begingroup$ No, the $X_i$'s are identically distributed under stationarity, all with the same distribution $\pi$ but they are dependent from one another. They are thus "did" rather than "iid". $\endgroup$ – Xi'an Jan 14 at 10:27

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