Difference between revisions of "Branching process, age-dependent"
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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 | + | 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 | + | 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==== | ||
| − | + | {| | |
| − | + | |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]]. | ||
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.
Branching process, age-dependent. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Branching_process,_age-dependent&oldid=25672