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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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There must not be eigenvalues $\lambda$ with $|\lambda|\ge 1$ and $\lambda\ne1$. –  Hagen von Eitzen Dec 10 '12 at 16:35
    
@HagenvonEitzen: Thanks! do you mean all eigenvalues of transition matrix have absolute value strictly less than 1? Why is that? –  Tim Dec 10 '12 at 16:37
    
@HagenvonEitzen I think that the eigenvalues of a stochastic matrix can not exceed one by the Perron-Frobenius theorem en.wikipedia.org/wiki/Stochastic_matrix –  Learner Dec 10 '12 at 16:44
    
@Learner OK, so we can simply say: There must not be eigenvalues $\lambda$ with $|\lambda|=1$ and $\lambda\ne 1$. –  Hagen von Eitzen Dec 10 '12 at 16:48
    
@Tim Yes, e.g. Michael Hardy's example is a case where an eigenvalue $-1$ occurs. –  Hagen von Eitzen Dec 10 '12 at 16:50
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1 Answer

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. –  Tim Dec 10 '12 at 16:39
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