Markov Chains | Part IB, 2004

Let (Xr)r0\left(X_{r}\right)_{r \geqslant 0} be an irreducible, positive-recurrent Markov chain on the state space SS with transition matrix (Pij)\left(P_{i j}\right) and initial distribution P(X0=i)=πi,iSP\left(X_{0}=i\right)=\pi_{i}, i \in S, where (πi)\left(\pi_{i}\right) is the unique invariant distribution. What does it mean to say that the Markov chain is reversible?

Prove that the Markov chain is reversible if and only if πiPij=πjPji\pi_{i} P_{i j}=\pi_{j} P_{j i} for all i,jSi, j \in S.

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