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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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Why the interest? – Did Jul 23 '13 at 20:39
@Did Chains satisfying these properties have interesting relations to some machine learning problems. – Vedarun Jul 25 '13 at 18:17
Any reference about these? – Did Jul 25 '13 at 20:08
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. – Vedarun Jul 26 '13 at 6:27

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