=+20.7. Let X0, X1, ... be a persistent, irreducible Markov chain, and for a fixed state j let T1, T2, ... be the times of the successive passages through j. Let Z, = T, and Z ,, = T ., - T ,, - 1, n ≥ 2. Show that Z1, Z2, ... are independent and that P[Z„ = k] = f(*) for n ≥ 2.