Galton-Watson Process

A Galton-Watson process is a branching process $\{Z_n\}$ with $Z_0=1$ (typically), where each individual in generation $n$ independently produces offspring according to a common probability distribution $\{p_k{\}}_{k\ge 0}$ (the same distribution for every individual, in every generation), so that $Z_{n+1}={\sum }_{i=1}^{Z_n}{\xi }_{n,i}$, with ${\xi }_{n,i}$ i.i.d. draws from $\{p_k\}$. Letting $\varphi (s)={\sum }_k{p}_k{s}^k$ denote the offspring distribution's probability generating function, the extinction probability $q=P(Z_n=0\text{ for some }n)$ is the smallest solution in $[0,1]$ of the fixed-point equation $q=\varphi (q)$. If the mean offspring number $\mu ={\varphi }^{\prime }(1)\le 1$ (subcritical or critical), then $q=1$, extinction is certain; if $\mu >1$ (supercritical), then $q<1$, and the process survives with positive probability $1-q$. On the event of survival, a supercritical Galton-Watson process grows geometrically, $Z_n/{\mu }^n$ converges almost surely to a non-degenerate random limit (a consequence of Doob's martingale convergence theorem, since $Z_n/{\mu }^n$ is a martingale).

• Originally motivated by an 1873 puzzle about why aristocratic English surnames kept vanishing over generations, the Galton-Watson process's extinction analysis (via probability generating functions) remains, over 150 years later, the standard textbook technique for computing exact extinction probabilities.
• The extinction-probability fixed-point equation $q=\varphi (q)$ always has $q=1$ as one solution (since $\varphi (1)=1$ for any valid probability generating function), the real content of the theorem is identifying when a smaller root also exists, exactly when the mean offspring number exceeds 1.
• Beyond surname genealogy, the same mathematics underlies epidemic outbreak extinction probabilities (will an infection chain die out, given each case's number of secondary infections?) and nuclear chain reaction criticality, both direct real-world reinterpretations of the same subcritical/critical/supercritical trichotomy.