Probability theory and stochastic processes Definition

Galton–Watson process

Builds on Independent and identically distributed random variables Stochastic process Probability-generating function Markov chain

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 $m > 1$ .

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Galton–Watson process definition

Tag	`0H631CZP5K`
URI	`m2://0H631CZP5K`
Kind	definition
Writing	W0 stub
MSC	60J80
Defines	Galton–Watson process
Uses	Independent and identically distributed random variables, Stochastic process, Probability-generating function, Markov chain

Galton–Watson process

Definition

Remarks

Reading path

Further reading

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