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In a Markov chain (you can add additional conditions here, such as discrete-time, homogeneous, finite-state, .... But the less additional condition, the better ), what sufficient and/or necessary condition can make every initial distribution have a limit distribution?

Note that here the limit distributions for different initial distributions may be different. Added: What I was thinking when posting the question is to include the case when there does not exist the limiting distribution same for all initial distributions, but there exists a limit distribution for every initial distribution.

Thanks and regards!

My question comes from my comment to Michael Hardy's reply.

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    @Tim Consider the Jordan normal form. Block belonging to $\lambda=1$ correspond to limit distributions, parts belonging to blocks with |\lambda|<1 will simply §fade away" in the long rund, but vectors corrsponidng to other blocks with $|\lambda|=1$ keep oscillating.2012-12-10

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I think that for a finite-state discrete-time Markov chain, sufficient conditions are that the chain be irreducible aperiodic and positive recurrent. In that case, it will be ergodic and will possess a unique limiting distribution.

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    +1 Thanks! I did know that. What I was thinking when posting the question is to include the case when there does not exist the limiting distribution same for all initial distributions, but there exist a limit distribution for every initial distribution.2012-12-10