38. Recall that state i is said to be positive recurrent if mi,i < ∞, where mi,i is the expected number of transitions until the Markov chain, starting in state i, makes a transition back into that state. Because πi, the long run proportion of time the Markov chain, starting in state i, spends in state i, satisfies

πi = 1 mi,i it follows that state i is positive recurrent if and only if πi > 0. Suppose that state i is positive recurrent and that state i communicates with state j . Show that state j is also positive recurrent by arguing that there is an integer n such that

πj πiPn i,j > 0