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

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

$$G_{n}(s)=\frac{n-(n-1) s}{n+1-n s} \quad \text { for }|s|<1+\frac{1}{n} .$$

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

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

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

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