Paper 2, Section II, F

Probability | Part IA, 2012

(i) Let XnX_{n} be the size of the nth n^{\text {th }}generation of a branching process with familysize probability generating function G(s)G(s), and let X0=1X_{0}=1. Show that the probability generating function Gn(s)G_{n}(s) of XnX_{n} satisfies Gn+1(s)=G(Gn(s))G_{n+1}(s)=G\left(G_{n}(s)\right) for n0n \geqslant 0.

(ii) Suppose the family-size mass function is P(X1=k)=2k1,k=0,1,2,P\left(X_{1}=k\right)=2^{-k-1}, k=0,1,2, \ldots Find G(s)G(s), and show that

Gn(s)=n(n1)sn+1ns for s<1+1n.G_{n}(s)=\frac{n-(n-1) s}{n+1-n s} \quad \text { for }|s|<1+\frac{1}{n} .

Deduce the value of P(Xn=0)P\left(X_{n}=0\right).

(iii) Write down the moment generating function of Xn/nX_{n} / n. Hence or otherwise show that, for x0x \geqslant 0,

P(Xn/n>xXn>0)ex as nP\left(X_{n} / n>x \mid X_{n}>0\right) \rightarrow e^{-x} \quad \text { as } n \rightarrow \infty

[You may use the continuity theorem but, if so, should give a clear statement of it.]

Typos? Please submit corrections to this page on GitHub.