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The reversibility of a Markov chain is defined in the following way with some basic propositions.

Unfortunately all examples of reversible Markov chains shown in my textbook so far are irreducible, giving me an impression (I think it is false) that all reversible Markov chains are irreducible.

I am curious about is there any example of reversible yet reducible Markov chains, so the initial distribution satisfying the detailed balance is not the only stationary distribution. Thank you!

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If the chain has an absorbing state, then the answer is no, because then $P(X_0=x_0,...X_n=x_n)>0$ but $P(X_n=x_n,...,X_0=x_0)=0$, where $x_n$ is the absorbing state. On the other hand, suppose your markov chain can be broken down into two disjoint communicating classes, i.e. two disconnected Markov chains. Then if the individual markov chains are reversable, then so is the full chain, since it's impossible to cross from one to the other.

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