Galton-Watson Process
Definition 1
The Basic Reproductive Rate $m = EX < \infty$ of a random variable $X$ is given. If we denote the $X_{n,i}$th offspring of the $i$th particle in the $n$rd generation of a branching process by $X_{n,i}$, then the stochastic process represented by a random sample $\left\{ X_{n,i} : (n,i) \in \mathbb{N}^{2} \right\} \overset{\text{iid}}{\sim} X$ is called the Galton-Watson Process. $$ Z_{n+1} = \sum_{i=1}^{Z_{n}} X_{n,i} $$ If the mean $m$ of $X$ is $m=1$, it’s called Critical, if $m > 1$ then Supercritical, if $m > 1$ then Subcritical. When all particles disappear, i.e., $Z_{n} = 0$, it’s referred to as Extinction.
Extinction Theorem 1
Let’s consider $Z_{0} = N \in \mathbb{N}$ and the probability generating function $f_{X}(s) = E s^{X} = \sum_{k=0}^{\infty} s^{k} P \left( X = k \right)$.
- If it’s not supercritical, then it ultimately becomes extinct. In other words, the following holds. $$ \lim_{n \to \infty} P \left( Z_{n} = 0 \right) = 0 $$
- If it’s supercritical, a unique $q \in (0,1)$ that satisfies $f(q) = q$ exists, and the extinction probability is as follows. $$ \lim_{n \to \infty} P \left( Z_{n} = 0 \right) = q^{N} $$
Explanation
The Galton-Watson process is mathematically the neatest, historically the oldest, simplest, and most famous branching process. Watson and Galton first devised and applied the branching process in their study on the diffusion of a particular surname.
The Galton-Watson processing introduced in the definition refers to a simple case, meaning the population is limited to $X$. However, this type can be extended, and the random variable can be turned into a random vector to generalize into a Multiple Galton-Watson Process. This allows for the construction of gender and age considerations or nondeterministic SIR models, among others.
In the extinction theorem, the formula for the supercritical case can be interpreted frequentistically, suggesting that for a sufficiently large $n$, by observing the realization $z_{n}$ of $Z_{n}$, it can be assumed that approximately $q^{N}\%$ of the total will become extinct, and the rest will not. Of course, if it’s not supercritical, everything ultimately becomes extinct. For example, if $q^{N} = 0.002$, it’s expected to become extinct about twice in $1000$ simulations.