Bienaymé–Galton–Watson Simple Branching Process and Extinction

Abstract

The Bienaymé–Galton–Watson simple branching process is defined by the successive numbers X _n of progeny at the n-th generation, n = 0, 1, 2, …, recursively and independently generated according to a given offspring distribution, starting from a non-negative integer number of initial X ₀ progenitors. The state zero, referred to as extinction, is an absorbing state for the process. In this chapter a celebrated formula for the probability of extinction is given as a fixed point of the moment generating function of the offspring distribution. The mean μ of the offspring distribution is observed to play a characteristic role in the determination of the behavior of the generation sizes X _n as n →∞. The critical case in which μ = 1 is analyzed under a finite second moment condition to determine the precise asymptotic nature of the survival probability, both unconditionally and conditionally on survival, in a theorem referred to as the Kolmogorov–Yaglom–Kesten–Ney–Spitzer theorem.