GaltonWatsonvertakkingsproces

The Galton–Watson process, also known as a Galton–Watson branching process, is a discrete-time stochastic model of population growth. It describes the number of individuals in successive generations, where each individual in generation n independently produces a random number X of offspring according to a fixed distribution, and all individuals act independently. If Z_n denotes the population size in generation n, then Z_{n+1} = sum_{i=1}^{Z_n} X_i with Z_0 given.

Let m = E[X] be the mean number of offspring per individual. If m > 1 the process is supercritical, if m = 1 critical, and if m < 1 subcritical. In all cases Z_n forms a Markov chain on the nonnegative integers, with E[Z_{n+1} | Z_n] = m Z_n and E[Z_n] = m^n Z_0.

Extinction, the event that Z_n = 0 for some n, occurs with probability q, which is the smallest nonnegative root of the generating function equation q = G(q), where G(s) = E[s^X] is the offspring generating function. For m <= 1, q = 1; for m > 1, q < 1, so survival is possible.

Variants: the multitype Galton–Watson process extends to offspring types; continuous-time versions and general branching processes generalize the framework. Applications include biology, genetics, ecology, epidemiology, and the analysis of family-name extinction; it also informs algorithms and radiation transport models.

History: named after Francis Galton and William Watson, who studied surname extinction; formal development occurred in the 20th century, with key treatments by Harris and later Athreya and Ney.