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Let $P$ be the transition probability matrix of a aperiodic irreducible DTMC and let $\pi$ be its stationary distribution. I would like to know if there is any literature on types of Markov chains that satisfy the following clause:

if $P_{ij} \ge P_{ji}$, then $\pi_i \le \pi_j$ for all $i,j$

I understand the time reversible chains are a special case of the above type, but in general what other properties can be said about these chains?

Thanks in advance.

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  • $\begingroup$ Why the interest? $\endgroup$
    – Did
    Commented Jul 23, 2013 at 20:39
  • $\begingroup$ @Did Chains satisfying these properties have interesting relations to some machine learning problems. $\endgroup$
    – Vedarun
    Commented Jul 25, 2013 at 18:17
  • $\begingroup$ Any reference about these? $\endgroup$
    – Did
    Commented Jul 25, 2013 at 20:08
  • $\begingroup$ That's what i am trying to figure out myself! I am currently working on a paper where such chains seem to be of use (think of Google page rank chains). I was wondering if these chains must necessarily be time reversible. Probably not, but then i am not sure what other interesting characteristics these chains will have. $\endgroup$
    – Vedarun
    Commented Jul 26, 2013 at 6:27

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