View source for Branching process, age-dependent

{{MSC|60J80}}

[[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

<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>

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

<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>

where

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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>

<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>

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.

<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 {{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

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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" />:

<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>

<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 {{Cite|S2}}, {{Cite|M}}.

====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====
Additional references can be found in the article [[Branching process|Branching process]].