Let $X=\{X_0,X_1,\ldots\}$ be an irreducible, positive recurrent, and aperiodic Markov chain with the state space $S=\{0,1,2,\ldots\}$ then how do we show that the probability $$ \lim_{n\rightarrow\infty}P(X_n=k\mid X_0=j) $$ is independent of $j$ and strictly positive? That is, why $$ \lim_{n\rightarrow\infty}P(X_n=k\mid X_0=j)=g(k)>0? $$

  • $\begingroup$ You need the Markov chain to be irreducible and aperiodic, not just positive recurrent. $\endgroup$ – Robert Israel Sep 28 '14 at 22:24
  • $\begingroup$ @ Robert: You are absolutely right. I forgot to mention that in my question but now I add this assumption too. Thanks! $\endgroup$ – user176667 Sep 28 '14 at 22:25
  • $\begingroup$ @Inj You really need it to be aperiodic as well, otherwise the limit may not exist. $\endgroup$ – Sasha Sep 28 '14 at 22:33
  • $\begingroup$ @ Sasha: Yes, that's true. $\endgroup$ – user176667 Sep 28 '14 at 22:34

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