General birth and death chains. The state space is f0; 1; 2; : : :g and the transition probability has p.x; x C 1/ D px p.x; x  1/ D qx forx > 0 p.x; x/ D 1  px  qx for x  0 while the other p.x; y/ D 0. Let Vy D minfn  0 W Xn D yg be the time of the first visit to y and let hN .x/ D Px.VN < V0/. Let .z/ D Pz yD1 Qy1 xD1 qx=px.

Show that Px.Vb < Va/ D

.x/  .a/

.b/  .a/

From this it follows that 0 is recurrent if and only if .b/!1as b !1, giving another solution of 1.70 from Chap. 1.