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A Markov chain with transition matrix $P$ is termed regular if for some $n$, all entries of $P^n$ are positive.

Is there a known notion of degree of regularity quantified in terms of how soon all entries are positive? That is, a Markov chain with transition matrix $P$ would be considered more regular than a Markov chain with transition matrix $Q$ if $P^n$ has positive entries but $Q^m$ only has positive entries for $m>n$.

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    $\begingroup$ This would precisely capture the length of the longest path between all shortest paths in the markov chain graph $\endgroup$
    – nir shahar
    Commented Apr 1, 2021 at 1:19
  • $\begingroup$ Sounds like you've just defined such a notion. Is there something you want to know about it? $\endgroup$
    – D.W. ♦
    Commented Apr 1, 2021 at 6:27
  • $\begingroup$ @D.W. Yes I suppose so. I’m wondering if this notion has already been established in the literature. I’d like to read up on it if so. $\endgroup$
    – Erik M
    Commented Apr 1, 2021 at 6:28
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    $\begingroup$ If im not mistaken this is the diameter of the graph representing the merkov chain $\endgroup$
    – nir shahar
    Commented Apr 1, 2021 at 10:16
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    $\begingroup$ This would be something well known in the literature of graphs, but i don't know how useful it is in the literature of markov chains $\endgroup$
    – nir shahar
    Commented Apr 1, 2021 at 10:18

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