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Regular Transition Matrix A transition matrix $P$ is said to be regular if some power of $P$ is positive. That is, $P^n >0$,for some $n≥1$.

My question is: Do all regular matrices have limiting distributions? And do all matrices with limiting distributions imply that they are regular?

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Since the Markov chain is regular, there exists a $k$ such that for all states $i,j$ in the state space $P^k(i,j) > 0$. Naturally, a regular Markov chain is then irreducible.

Now, since $P^k > 0$ for all $i,j$, then $P^{k+1}$ has entries $$\pi_{ij}^{k+1} = \sum_t \pi_{it} \pi_{tj}^{k} > 0 \,, $$ since for at least one $t$, $\pi_{it} > 0$. This means that $P^{k+1}$ > 0 as well, and so is $P^{k+2}, P^{k+3},...$

The gcd of $\{k, k+1, k+2, \dots\}$ is 1, and thus a regular Markov chain is also aperiodic. Thus, regular Markov chains are irreducible and aperiodic which implies, the Markov chain has a unique limiting distribution.

Conversely, all matrices with a limiting distribution do not imply that they are regular. A counter-example is the example here, where the transition matrix is upper triangular, and thus the transition matrix for every step is upper triangular (and hence not regular), but a limiting distribution exists.

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    $\begingroup$ +1. It would be good to draw the conclusion explicitly: irreducible and aperiodic imply a unique limiting distribution. $\endgroup$ – whuber Feb 7 '18 at 14:27

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