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<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>
 
<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>
  
The following results have been obtained for these processes [[#References|[1]]]: 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
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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
  
 
<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>
 
<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>
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<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>
 
<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>
  
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 [[#References|[2]]], [[#References|[3]]].
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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}}.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> B.A. [B.A. Sevast'yanov] Sewastjanow, "Verzweigungsprozesse" , Akad. Wissenschaft. DDR (1974) (Translated from Russian) {{MR|0408018}} {{ZBL|0291.60039}} </TD></TR>
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{|
<TR><TD valign="top">[2]</TD> <TD valign="top"> 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 {{MR|0170396}} {{ZBL|0248.60059}} </TD></TR>
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|valign="top"|{{Ref|S}}|| B.A. Sewastjanow, "Verzweigungsprozesse" , Akad. Wissenschaft. DDR (1974) (Translated from Russian) {{MR|0408018}} {{ZBL|0291.60039}}
<TR><TD valign="top">[3]</TD> <TD valign="top"> C.J. Mode, "Multitype branching processes" , Elsevier (1971) {{MR|0279901}} {{ZBL|0219.60061}} </TD></TR></table>
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|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}}
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|valign="top"|{{Ref|M}}|| C.J. Mode, "Multitype branching processes" , Elsevier (1971) {{MR|0279901}} {{ZBL|0219.60061}}
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====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]].

Revision as of 06:17, 11 May 2012

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 [S]: 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 [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