# 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 [1]: 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 [2], [3].

#### References

 [1] B.A. [B.A. Sevast'yanov] Sewastjanow, "Verzweigungsprozesse" , Akad. Wissenschaft. DDR (1974) (Translated from Russian) [2] B.A. Sevast'yanov, "Age-dependent branching processes" Theory Probab. Appl. , 9 : 4 (1964) pp. 521–537 Teor. Veroyatnost. i Primenen. , 9 : 4 (1964) pp. 577–594 [3] C.J. Mode, "Multitype branching processes" , Elsevier (1971)