Difference between revisions of "Branching process, age-dependent"
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{{MSC|60J80}} | {{MSC|60J80}} | ||
[[Category:Branching processes]] | [[Category:Branching processes]] | ||
| − | 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 | + | 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 | + | 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 | where | ||
| − | + | $$ | |
| + | h (u, x) = \ | ||
| + | \sum _ {k = 0 } ^ \infty | ||
| + | p _ {k} (u) x ^ {k} . | ||
| + | $$ | ||
Put | 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 | + | 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 | + | The following results have been obtained for these processes {{Cite|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 | + | 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 | + | If $ h(u, x) $ |
| + | is independent of $ u $, | ||
| + | the age-dependent branching process is a [[Bellman–Harris process|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 {{Cite|S2}}, {{Cite|M}}. | ||
====References==== | ====References==== | ||
| − | + | {| | |
| − | + | |valign="top"|{{Ref|S}}|| B.A. Sewastjanow, "Verzweigungsprozesse" , Akad. Wissenschaft. DDR (1974) (Translated from Russian) {{MR|0408018}} {{ZBL|0291.60039}} | |
| − | + | |- | |
| + | |valign="top"|{{Ref|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 {{MR|0170396}} {{ZBL|0248.60059}} | ||
| + | |- | ||
| + | |valign="top"|{{Ref|M}}|| C.J. Mode, "Multitype branching processes" , Elsevier (1971) {{MR|0279901}} {{ZBL|0219.60061}} | ||
| + | |} | ||
====Comments==== | ====Comments==== | ||
Additional references can be found in the article [[Branching process|Branching process]]. | Additional references can be found in the article [[Branching process|Branching process]]. | ||
Latest revision as of 06:29, 30 May 2020
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 |
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