The following image describes a Galton-Watson process.
Let $\xi_i^n$, $i,n\geq1$, be i.i.d. nonnegative integer-valued random variables. Define a sequence $Z_n$, $n\geq0$ by $Z_0=1$ and $$Z_{n+1}=\left\{\begin{array}{lll} \xi_1^{n+1}+\cdots+\xi_{Z_n}^{n+1} & \text{if} & Z_n>0,\\ 0 & \text{if} & Z_n=0. \end{array}\right.$$
This is the Galton-Watson Process where the $Z_n$ term is supposed to represent the number of individuals in the $n^{th}$ generation.
If $\mu:=E[\xi_i^m]<1$ then $Z_n$ is $0$ a.s. for $n$ large enough. However, I do not understand the proof which goes as follows:
- $\frac{Z_n}{\mu^n}$ is a martingale. (This is fine.)
- Since $E[\frac{Z_n}{\mu^n}]=E[X_0]=1$ this implies that $E[Z^n]=\mu^n$. Also $Z_n1_{\{Z_n>0\}}\geq 1_{\{Z_n>0\}}$ which implies that:$$P(Z_n>0)\leq E[Z_n;Z_n>0]\leq E[Z_n]=\mu^n\rightarrow 0.$$
The proof ends there. However, I don't see why this shows that $Z_n=0$ for large enough $n$. It just shows that for large enough $n$, $P(Z_n>0)$ is really small but it might not be zero which is what we want since almost surely requires $P(Z_n=0)=1$ for large enough $n$.
Any ideas how one might complete the proof?
$Z_n=0$ implies $Z_{n+1}=0$. Let $E_n$ be the event $Z_n=0$. Then $P(Z_n=0 \text { for some } n)=P(\bigcup_n E_n)=\lim_{n \to \infty} P(E_n)$ since $E_n$'s are increasing. Hence,$P(Z_n=0 \text { for some } n)=1$.