=+8.38. 5.121 Consider an irreducible, aperiodic, positive persistent chain. Let T, be the smallest n such that X ,, = j, and let m ;, = E,[7;]. Show that there is an r such that p= PiX, + j. ..., X, -, + j, X, = i] is positive; from ff"+") > pf (") and m, < co, conclude that m .; < > and my = > ;- P.(+, > n]. Starting from p =

_4 _. f("p"; - $), show that C (p) - pp) = 1 - C pl1-1)P[7]> m].

m-0 Use the M-test to show that TTm = 1 + 2 (pl) - p;).

n-1

= If ij, this gives m, 1/m, again; if i +j, it shows how in principle m,, can be calculated from the transition matrix and the stationary probabilities.