91Ó°ÊÓ

For a time reversible Markov chain, argue that the rate at which transitions from \(i\) to \(j\) to \(k\) occur must equal the rate at which transitions from \(k\) to \(j\) to \(i\) occur.

A Markov chain is said to be a tree process if (i) \(P_{i j}>0\) whenever \(P_{j i}>0\). (ii) for every pair of states \(i\) and \(j, i \neq j\), there is a unique sequence of distinct states \(i=i_{0}, i_{1}, \ldots, i_{n-1}, i_{n}=j\) such that $$ P_{i_{k}, i_{k+1}}>0, \quad k=0,1, \ldots, n-1 $$ In other words, a Markov chain is a tree process if for every pair of distinct states \(i\) and \(j\) there is a unique way for the process to go from \(i\) to \(j\) without reentering a state (and this path is the reverse of the unique path from \(j\) to \(i\) ). Argue that an ergodic tree process is time reversible.