Let (Sn)n∈N be a random walk process with increments that are independent. The value of the random walk increase by one in one time step with probability p and decrease by one in one time step with probability q = 1 − p. We denote the hitting time of state 0 to be T0 = inf{n ≥ 0 : Sn = 0}.
(a) Show that for any i ≥ 1, we have E[T0|S0 = i] = iE[T0|S0 = 1] , and compute E[T0|S0 = 1] for the case q > p and p ≥ q.
(b) Similarly, show that we have, P(T0 < ∞|S0 = i) = (P(T0 < ∞|S0 = 1))^i , i ≥ 1.
(c) Find P(T0 < ∞|S0 = 1).