GaltonWatson

GaltonWatson, commonly called the Galton–Watson process, is a discrete-time stochastic process that models a population where each individual independently gives birth to a random number of offspring according to a fixed distribution and then dies. It was introduced by Francis Galton and Henry William Watson in the 19th century in their study of whether a given family name would persist over generations.

Definition and formulation: Let Z_0 = 1 denote the initial ancestor, and conditional on Z_n, Z_{n+1} = sum_{i=1}^{Z_n} X_i, where the X_i are independent and identically distributed nonnegative integer-valued random variables with P(X_i = k) = p_k. The offspring distribution has generating function f(s) = sum_{k>=0} p_k s^k, with mean m = f'(1) = E[X_i].

Extinction and growth: The extinction probability q is the smallest nonnegative solution to f(q) = q in the interval [0, 1]. If m <= 1, then q = 1, so extinction occurs almost surely; if m > 1, then q < 1 and the process survives with probability 1 − q. In expectation, E[Z_n] = m^n. The process is subcritical if m < 1, critical if m = 1, and supercritical if m > 1.

Variants and impact: The Galton–Watson framework has numerous variants, including multitype versions and continuous-time branching processes. It serves as a foundational model in population genetics, ecology, epidemiology, and the analysis of branching algorithms, illustrating how simple local reproduction rules can lead to complex, probabilistic population dynamics.