Solve the following:

A branching process Z = (Zn)no is defined recursively by a given family of non-negative integer valued iid. random variables { X(") as k,n=1 follows: Zo : = 1, Zn+1 : = X, "+1 ) + ... + x(ntl), n>0. Let M = E X,") and Fn = (Zo, Z1, . .. Zn). We write f(s) for the generating function of the offspring distribution, that is f ( s ) = E (s* " ) - EP ( x ) = m sm m = 0 for any k and n. Further, let {extinction } = {Zn -+ 0} = {n, Zn =0}, {explosion} = {Zn -> co} and denote q = P (extinction). Recall that q is the smaller (if there are two) fixed point of f(s), that is q is the smallest solution of f (q) = q. (a) Prove by induction that E [Zn] = un. (b) Show that E [santi|Fr] = f(s)Zn for every s 2 0. Explain that qun is a martingale and lim Zn = Zoo exists a.s. (c) Let T = min {n : Zn = 0} be the extinction time with T = 0% if Zn > 0 always. Prove by dominated convergence that q = E [qZT] = E [q7. . Ir-] + E [q r . Ir