# Branching process, age-dependent

2020 Mathematics Subject Classification: Primary: 60J80 [MSN][ZBL]

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 $\tau$ with distribution function

$${\mathsf P} \{ \tau \leq t \} = G (t).$$

At the end of its life the particle is transformed into $k$ daughter particles of age zero with a probability $p _ {k} (u)$ if the transformation took place when the age attained by the original particle was $u$. Let $\mu (t)$ be the number of particles at the moment of time $t$. The generating function $F(t; x)$ of the probability distribution of $\mu (t)$ for a process beginning with one particle of age zero satisfies the equation

$$\tag{* } F (t; x) = \ \int\limits _ { 0 } ^ { t } h (u, F (t - u ; x)) dG (u) + x (1 - G (t)),$$

where

$$h (u, x) = \ \sum _ {k = 0 } ^ \infty p _ {k} (u) x ^ {k} .$$

Put

$$\left . a (u) = \ \frac{\partial h }{\partial x } \ \right | _ {x = 1 } ,\ \ \left . b (u) = \ \frac{\partial ^ {2} h }{\partial x ^ {2} } \ \right | _ {x = 1 } ,$$

$$A = \int\limits _ { 0 } ^ \infty a (u) dG (u),\ B = \int\limits _ { 0 } ^ \infty b (u) dG (u).$$

An age-dependent branching process is said to be subcritical, critical or supercritical if $A > 1$, $A = 1$ and $B > 0$, or $A > 1$, respectively. The behaviour of the process as $t \rightarrow \infty$ substantially depends on its criticality. Subcritical and critical processes die out with probability one, i.e.

$$\lim\limits _ {t \rightarrow \infty } \ {\mathsf P} \{ \mu (t) = 0 \} = 1.$$

The following results have been obtained for these processes [S]: asymptotic formulas for the moments $\mu (t)$, necessary and sufficient conditions of extinction, conditions of existence and uniqueness of a solution of equation (*) and asymptotic formulas as $t \rightarrow \infty$ for

$$Q (t) = \ {\mathsf P} \{ \mu (t) > 0 \} .$$

The limit distributions have also been determined. In the critical case, as $t \rightarrow \infty$:

$$Q(t) \approx \ 2 \frac{\int\limits _ { 0 } ^ \infty u a (u) dG (u) }{Bt } ,$$

$${\mathsf P} \left \{ \frac{\mu (t) }{ {\mathsf E} ( \mu (t) \mid \mu (t) > 0) } < x \mid \mu (t) > 0 \right \} \rightarrow 1 - e ^ {-x} ,\ x > 0.$$

If $h(u, x)$ is independent of $u$, 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 [S2], [M].

#### References

 [S] B.A. Sewastjanow, "Verzweigungsprozesse" , Akad. Wissenschaft. DDR (1974) (Translated from Russian) MR0408018 Zbl 0291.60039 [S2] B.A. Sewastjanow, "Age-dependent branching processes" Theory Probab. Appl. , 9 : 4 (1964) pp. 521–537 Teor. Veroyatnost. i Primenen. , 9 : 4 (1964) pp. 577–594 MR0170396 Zbl 0248.60059 [M] C.J. Mode, "Multitype branching processes" , Elsevier (1971) MR0279901 Zbl 0219.60061