A stochastic process where individuals have offspring. This might be modelled by having each individual in a generation of size Zn having offspring identically, independently distributed with pgf G(s). The pgf of Zn is then Gn(s), where Gn denotes n applications of G. The probability that the population becomes extinct equals the smallest non-negative solution of the equation s = G(s).
A stochastic process with varying numbers of states. Consider a population in which each individual has probability pk of providing k descendants in the next generation. Let Xj be the number of individuals in the jth generation. The sequence X1, X2,…, is a Markov process known as a branching process. This formulation was originally suggested by Galton and Henry Watson in 1874 in the context of the extinction of surnames, and may be referred to as the Galton–Watson model. Extinction is certain if and only if
and otherwise the probability of extinction is equal to P, the solution (between 0 and 1) of
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- Comnena, Anna (1083–1148)
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- compactness