Galton–Watson process

Definition

Let $\mu$ be a probability distribution on $\{0, 1, 2, \dots\}$, the offspring distribution. Let $(\xi_{n, i})_{n \ge 0, \, i \ge 1}$ be i.i.d. random variables with law $\mu$. The Galton–Watson process with offspring distribution $\mu$ is the stochastic process $(Z_n)_{n \ge 0}$ with $$Z_0 = 1 \quad \text{and} \quad Z_{n+1} = \sum_{i=1}^{Z_n} \xi_{n, i} \quad \text{for } n \ge 0.$$ Here $Z_n$ is the size of generation $n$, and an empty sum is $0$.

Remarks

$(Z_n)$ is a Markov chain on $\{0, 1, 2, \dots\}$, and $0$ is an absorbing state. Extinction is the event that $Z_n = 0$ for some $n$. Let $m$ be the mean of $\mu$ and let $G$ be its probability-generating function. Then $E[Z_n] = m^n$, and the generating function of $Z_n$ is the $n$-fold composition $G \circ \dots \circ G$. The extinction probability is the smallest solution $q \in [0, 1]$ of $G(q) = q$. Assume $\mu(\{1\}) < 1$. Then $q = 1$ if $m \le 1$, and $q < 1$ if $m > 1$.

Explanations

L1 human

The Galton–Watson process follows a family tree in which each person has a random number of children. If people have on average at most one child, the family dies out with probability $1$. If they have more, it can survive forever.