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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017540/b0175401.png" /> with distribution function
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017540/b0175402.png" /></td> </tr></table>
+
$$
 +
{\mathsf P} \{ \tau \leq  t \}  = G (t).
 +
$$
  
At the end of its life the particle is transformed into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017540/b0175403.png" /> daughter particles of age zero with a probability <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017540/b0175404.png" /> if the transformation took place when the age attained by the original particle was <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017540/b0175405.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017540/b0175406.png" /> be the number of particles at the moment of time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017540/b0175407.png" />. The generating function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017540/b0175408.png" /> of the probability distribution of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017540/b0175409.png" /> for a process beginning with one particle of age zero satisfies the equation
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017540/b01754010.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
+
$$ \tag{* }
 +
F (t; x)  = \
 +
\int\limits _ { 0 } ^ { t }
 +
h (u, F  (t - u ; x))
 +
dG (u) + x (1 - G (t)),
 +
$$
  
 
where
 
where
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017540/b01754011.png" /></td> </tr></table>
+
$$
 +
h (u, x)  = \
 +
\sum _ {k = 0 } ^  \infty 
 +
p _ {k} (u) x  ^ {k} .
 +
$$
  
 
Put
 
Put
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017540/b01754012.png" /></td> </tr></table>
+
$$
 +
\left . a (u)  = \
 +
 
 +
\frac{\partial  h }{\partial  x }
 +
\
 +
\right | _ {x = 1 }  ,\ \
 +
\left . b (u)  = \
 +
 
 +
\frac{\partial  ^ {2} h }{\partial  x  ^ {2} }
 +
\
 +
\right | _ {x = 1 }  ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017540/b01754013.png" /></td> </tr></table>
+
$$
 +
= \int\limits _ { 0 } ^  \infty  a (u)  dG (u),\  B  = \int\limits _ { 0 } ^  \infty  b (u)  dG (u).
 +
$$
  
An age-dependent branching process is said to be subcritical, critical or supercritical if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017540/b01754014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017540/b01754015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017540/b01754016.png" />, or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017540/b01754017.png" />, respectively. The behaviour of the process as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017540/b01754018.png" /> substantially depends on its criticality. Subcritical and critical processes die out with probability one, i.e.
+
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.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017540/b01754019.png" /></td> </tr></table>
+
$$
 +
\lim\limits _ {t \rightarrow \infty } \
 +
{\mathsf P} \{ \mu (t) = 0 \}  = 1.
 +
$$
  
The following results have been obtained for these processes {{Cite|S}}: asymptotic formulas for the moments <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017540/b01754020.png" />, necessary and sufficient conditions of extinction, conditions of existence and uniqueness of a solution of equation (*) and asymptotic formulas as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017540/b01754021.png" /> for
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017540/b01754022.png" /></td> </tr></table>
+
$$
 +
Q (t)  = \
 +
{\mathsf P} \{ \mu (t) > 0 \} .
 +
$$
  
The limit distributions have also been determined. In the critical case, as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017540/b01754023.png" />:
+
The limit distributions have also been determined. In the critical case, as $  t \rightarrow \infty $:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017540/b01754024.png" /></td> </tr></table>
+
$$
 +
Q(t)  \approx \
 +
2
 +
\frac{\int\limits _ { 0 } ^  \infty  u a (u)  dG (u) }{Bt }
 +
,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017540/b01754025.png" /></td> </tr></table>
+
$$
 +
{\mathsf P} \left \{
 +
\frac{\mu (t) }{ {\mathsf E} ( \mu (t) \mid  \mu (t)
 +
> 0) }
 +
< x \mid  \mu (t) > 0 \right \}  \rightarrow  1 - e  ^ {-x} ,\  x > 0.
 +
$$
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017540/b01754026.png" /> is independent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017540/b01754027.png" />, 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}}.
+
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====

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.

How to Cite This Entry:
Branching process, age-dependent. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Branching_process,_age-dependent&oldid=26370
This article was adapted from an original article by V.P. Chistyakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article