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For a discrete time Markov chain, its limiting distribution is defined to be the same for all the initial distributions. A distribution over the state space is called a reversible distribution, if it satisfies the detailed balance equation.

I think a reversible distribution may not be the limiting distribution, because it may happen that a reversible distribution exists while the limiting distribution doesn't. But when the limiting distribution exist, all stationary distributions (including reversible distributions) must be the same as the limiting distribution. Please correct me if I was wrong.

Conversely, must the limiting distribution (if exists) be a reversible distribution? A counterexample will suffice. I can imagine that counterexample Markov chain will need to have the limiting distribution, and doesn't have a reversible distribution.

Thanks and regards!

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A limiting distribution may not be reversible. A reversible distribution is always a limiting distribution. A reversible distribution may not be unique. A limiting distribution may not be unique. (Four assertions, three basic blunders. Please find a book and study it! As already explained many times.) – Did Dec 1 '12 at 15:08
@did: By "the limiting distribution" I mean the limiting distribution which does not depend on the initial distributions. I thought it was unique. Which books do you recommend that I can find answers to most of my questions? – Tim Dec 1 '12 at 15:17
I am not sure the term reversible distribution makes sense because I think reversibility to be really a property of the markov process and not of a distribution per se. Isn't more accurate just to say a reversible process? – Learner Dec 1 '12 at 15:19
@learner: Thanks for the comment! For a reversible distribution in a homogeneous DTMC, check out my previous post…. – Tim Dec 1 '12 at 15:21
this question seems to be very related to that one. I would suggest you to check out the book "Markov Chains and Stochastic Stability" by Meyn and Tweedie (available online) to learn about the 'limiting' distributions. – S.D. Dec 2 '12 at 17:22

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