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For a Markov chain, I define a reversible distribution to be a distribution wrt which the MC is reversible to. A stationary distribution is defined as a distribution that once reached will stay. A reversible distribution is a stationary distribution. But not vice versa.

I was wondering if a MC that has a reversible distribution can have a stationary distribution which is not a reversible distribution?

My question comes from Wikipedia:

Let $X$ be a finite set and let $K(x,y)$ be the transition probability for a reversible Markov chain on $X$. Assume this chain has stationary distribution $\pi$.

$\pi$ seems to be used as a reversible distribution in the article. I was wondering why it doesn't say $\pi$ is a distribution wrt which the MC is reversible.

Thanks and regards!

Tim
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1 Answers1

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For an irreducible Markov chain, a stationary distribution exists if and only if all states are positive recurrent. When this condition is met, the stationary distribution is unique.

Thus, if a Markov chain (reversible or not) is irreducible and has a stationary distribution, this stationary distribution is unique.

If a Markov chain on a finite set is reversible in the sense that $\mu(x)K(x,y)=\mu(y)K(y,x)$ for every $x$ and $y$, for some measure $\mu$, then $\pi=\mu/|\mu|$ is a stationary distribution. Thus, if a Markov chain on a finite set is reversible and irreducible, it has exactly one stationary distribution, which is reversible.

Did
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  • +1 Thanks! So I guess that Wikipedia article should add: the finite-state reversible MC is additionally irreducible. Can I also guess for a general reversible MC, there might be a stationary distribution which is not reversible? – Tim Dec 13 '12 at 16:45
  • It seems to me that this does not answer the op questions. The question was if it is possible for a reversible Markov chain to posses a non-reversible stationary distribution. – Adam Jun 22 '15 at 09:59
  • @Adam Please read slowly: "if a Markov chain on a finite set is reversible and irreducible, it has exactly one stationary distribution, which is reversible." – Did Jun 22 '15 at 15:22
  • I am not saying that your answer is wrong, I am just saying that the op question is still open. – Adam Jun 22 '15 at 15:51
  • @Adam Yes. And I think this statement is false. – Did Jun 22 '15 at 15:51
  • It seems to me that you say if the MC is reversible and irreducible, then it can not posses a non-reversible stationary distribution because the stationary distribution is unique. But how can you assure that there is no reversible, reducible MC that posses a non reversible stationary distribution? – Adam Jun 22 '15 at 15:57
  • @Adam Because I can produce a stationary distribution and it happens to be reversible. – Did Jun 22 '15 at 15:59
  • Let us continue this discussion in chat. – Adam Jun 22 '15 at 16:00
  • @did: In your third paragraph you introduced the assumption that the Markov chain has a finite state space, which was absent from the original question. – MRule Dec 18 '23 at 10:12