# Branching process, age-dependent

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A model of a branching process in which the lifetime of a particle is an arbitrary non-negative random variable, while the number of daughter particles depends on its age at the moment of transformation. In the single-type particle model each particle has a random duration of life with distribution function At the end of its life the particle is transformed into daughter particles of age zero with a probability if the transformation took place when the age attained by the original particle was . Let be the number of particles at the moment of time . The generating function of the probability distribution of for a process beginning with one particle of age zero satisfies the equation (*)

where Put  An age-dependent branching process is said to be subcritical, critical or supercritical if , and , or , respectively. The behaviour of the process as substantially depends on its criticality. Subcritical and critical processes die out with probability one, i.e. The following results have been obtained for these processes : asymptotic formulas for the moments , necessary and sufficient conditions of extinction, conditions of existence and uniqueness of a solution of equation (*) and asymptotic formulas as for The limit distributions have also been determined. In the critical case, as :  If is independent of , the age-dependent branching process is a Bellman–Harris process. The model just described has been generalized to include processes with several types of particles, and also to processes for which a particle may generate new particles several times during its lifetime , .

How to Cite This Entry:
Branching process, age-dependent. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Branching_process,_age-dependent&oldid=13426
This article was adapted from an original article by V.P. Chistyakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article