# Degree of regularity of a Markov chain

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 $$m>n$$.

• This would precisely capture the length of the longest path between all shortest paths in the markov chain graph Apr 1 at 1:19
• Sounds like you've just defined such a notion. Is there something you want to know about it?
– D.W.
Apr 1 at 6:27
• @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. Apr 1 at 6:28
• If im not mistaken this is the diameter of the graph representing the merkov chain Apr 1 at 10:16
• 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 Apr 1 at 10:18