Section 15.3

Branching Processes

Extinction probabilities as fixed points of the offspring generating function.

Toolkit

A branching process $Z_n$ starts from $Z_0=1$ , each individual independently produces offspring with distribution $p_k=\mathbb{P}(\text{offspring}=k)$ and generating function $G(s)=\mathbb{E}[s^X]=\sum_k p_k s^k$ . Since $Z_{n+1}$ is a sum of $Z_n$ i.i.d. copies of $X$ , the generating function of $Z_n$ is the $n$ -fold composition $G_n(s)=G(G(\cdots G(s)))$ .

Key facts:

Extinction probability $q=\mathbb{P}(Z_n=0 \text{ for some } n)$ satisfies the fixed-point equation $q=G(q)$ , obtained by conditioning on the number of first-generation offspring.
$q$ is the smallest root of $G(s)=s$ in $[0,1]$ ; $s=1$ is always a root.
Let $\mu=G'(1)=\mathbb{E}[X]$ (mean offspring number). If $\mu\le 1$ (and $p_1\ne1$ ), then $q=1$ . If $\mu>1$ , there is a unique root $q\in(0,1)$ , giving $q<1$ .
$G$ is increasing and convex on $[0,1]$ , which guarantees existence/uniqueness of the relevant root and justifies “smallest root” via iterating $q_n=G(q_{n-1})$ , $q_0=0$ , increasing to $q$ .

Problems

15.8 Extinction of a Bacterial Colony+1 variation Medium Free