GaltonWatsonin

GaltonWatsonin is a probabilistic model of reproduction across generations, commonly known as the Galton–Watson branching process. It is a classical model in probability theory named after Francis Galton and the Reverend Henry William Watson, who studied the extinction of family names in the 19th century. The model describes a population where each individual in generation n produces a random number of offspring in generation n+1, independently of others, according to a fixed offspring distribution.

Formally, if X denotes the number of offspring produced by a single individual with distribution {p_k}, and Z_n is the number of individuals in generation n, then Z_{n+1} = sum_{i=1}^{Z_n} X_i^{(n)}, where the X_i^{(n)} are i.i.d. copies of X. The process starts with Z_0 initial individuals. A convenient tool is the generating function F(s) = E[s^X] = sum_k p_k s^k. The extinction probability q is the smallest nonnegative root of q = F(q). The mean offspring m = F'(1) determines the long-term behavior.

If m < 1 (subcritical) or m = 1 (critical), extinction occurs with probability 1, though the distribution of total progeny can vary. If m > 1 (supercritical), there is a positive probability of indefinite growth. Variants include multi-type Galton–Watson processes and continuous-time or environment-influenced versions. Applications span biology, epidemiology, genetics, nuclear physics, and the analysis of random trees in computer science. GaltonWatsonin remains a foundational concept linking generating functions, martingales, and stochastic growth.