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In the study of discrete-time Markov models, it is said that if a limiting distribution $\pi$ exists, then it satisfies $\pi_j = \sum_{i = 1}^N \pi_i p_{i, j}, \ j \in S$ and $\sum_{j = 1}^N \pi_j = 1$. Is the converse also true? That is, if $\pi_j = \sum_{i = 1}^N \pi_i p_{i, j}, \ j \in S$ and $\sum_{j = 1}^N \pi_j = 1$, then is $\pi$ a limiting distribution?

I would greatly appreciate it if people would please take the time to clarify this.

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    $\begingroup$ No. Even if we clarify that the pi are non-negative, there’s no reason for a limit to exist... the chain could be periodic. $\endgroup$ – user8675309 Mar 27 at 9:17
  • $\begingroup$ @user8675309 Thanks for the clarification. $\endgroup$ – The Pointer Mar 27 at 18:19

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