GaltonWatsonféle

The Galton-Watson process is a type of stochastic process that models population growth. It is named after Francis Galton and Henry William Watson, who first introduced it in 1874 to study the extinction of surnames. The process describes the evolution of a population where each individual in generation $n$ independently produces a random number of offspring for generation $n+1$. The offspring distribution, which specifies the probabilities of having $k$ offspring for any non-negative integer $k$, is assumed to be the same for all individuals in the population.

The state of the process at time $n$ is the number of individuals in the population at that generation, denoted by $Z_n$. The initial population size is $Z_0$. The core of the Galton-Watson process lies in the expected number of offspring produced by a single individual, known as the mean offspring number, often denoted by $m$. If $m < 1$, the population is expected to decline. If $m > 1$, the population is expected to grow. If $m = 1$, the population size is expected to remain stable, though extinction is still possible.

A key question in the study of Galton-Watson processes is the probability of extinction, which is the probability that the population size eventually becomes zero. This probability depends on the offspring distribution and the initial population size. The Galton-Watson process has applications in various fields, including biology (e.g., modeling cell division or gene extinction), genetics, and branching processes in general.