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 
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) MR0408018 Zbl 0291.60039 |
| [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 MR0170396 Zbl 0248.60059 |
| [3] | C.J. Mode, "Multitype branching processes" , Elsevier (1971) MR0279901 Zbl 0219.60061 |
Comments
Additional references can be found in the article Branching process.
Branching process, age-dependent. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Branching_process,_age-dependent&oldid=25671