BienayméGaltonWatson

The Bienaymé-Galton-Watson process is a discrete-time stochastic model used to describe the size of generations in a branching population. It is named for Valentin de Bienaymé and for Francis Galton and Sir Francis Watson, who together contributed to its development in the 19th century. The process is typically defined by a sequence Z0, Z1, Z2, … where Z0 is the initial population size (often 1). Given Zn, the next generation size Zn+1 is the sum of the offspring produced by each individual in generation n. If each individual produces offspring independently according to a common distribution {pk}, then Zn+1 = X1 + X2 + … + XZn, where the Xi are independent and identically distributed with P(Xi = k) = pk for k ≥ 0. The offspring distribution is encoded by its generating function f(s) = sum pk s^k.

Key properties include the mean m = E[Xi] = sum k pk. The value of m determines long-term behavior: if m ≤ 1 (subcritical or critical), extinction occurs with probability 1; if m > 1 (supercritical), there is a positive probability of non-extinction. The extinction probability q is the smallest nonnegative root of f(q) = q in the interval [0,1].

Historically, Bienaymé introduced branching concepts in 1845, and Galton and Watson popularized the model through studies of surname extinction and family lineages. The Bienaymé-Galton-Watson process remains a foundational framework in branching process theory with applications in biology, epidemiology, physics, and computer science. Extensions include multi-type and continuous-time branching processes, which generalize the basic idea to richer settings.