(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] 1/ D qx forx > 0 p.x; x/ D rx 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/. By considering what happens on the first step, we can write hN .x/ D pxhN .x C 1/ C rxhN .x/ C qxhN .x [1] 1/

Set hN .1/ D cN and solve this equation to conclude that 0 is recurrent if and only ifP 1 yD1Qy[1]1 xD1 qx=px D1where by conventionQ0 xD1 D 1.