---
tag: 0H631CZP5K
kind: definition
name: Galton–Watson process
msc: [60J80]
defines: 0G1022YXVQ
uses: [0AJN7WHBJZ, 0QH5ENKZM1, 0Z8PSSAVBA, 033TJBZDG0]
sources:
  - url: https://en.wikipedia.org/wiki/Galton–Watson_process
    locator: Mathematical definition
---

## 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.|0AJN7WHBJZ]] random variables with law $\mu$.
The Galton–Watson process with offspring distribution $\mu$ is the [[stochastic process|0QH5ENKZM1]] $(Z_n)_{n \ge 0}$ 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|033TJBZDG0]] 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|0Z8PSSAVBA]]. 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.