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Difference between revisions of "Branching process with immigration"

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(MSC|60J80 Category:Branching processes)
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[[Category:Branching processes]]
 
[[Category:Branching processes]]
  
A model of a branching process (discrete-time or continuous-time, with one or several types of particles, etc.) in which new particles may appear not only as the products of division of the original particles, but also as a result of immigration from some "external source" . For instance, let
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A model of a branching process (discrete-time or continuous-time, with one or several types of particles, etc.) in which new particles may appear not only as the products of division of the original particles, but also as a result of immigration from some "external source" . For instance, let
  
 
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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/b017580/b0175801.png" /></td> </tr></table>
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====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A.M. Zubkov,   "Life-like periods of a branching process with immigration" ''Theory Probab. Appl.'' , '''17''' : 1 (1972) pp. 174–183 ''Teor. Veroyatnost. i Primenen.'' , '''17''' : 1 (1972) pp. 179–188</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A.G. Pakes,   "Further results on the critical Galton–Watson process with immigration" ''J. Austral. Math. Soc.'' , '''13''' : 3 (1972) pp. 277–290</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> J.H. Foster,   J.A. Williamson,   "Limit theorems for the Galton–Watson process with time-dependent immigration" ''Z. Wahrsch. Verw. Geb.'' , '''20''' (1971) pp. 227–235</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> E. Seneta,   "An explicit limit theorem for the critical Galton–Watson process with immigration" ''J. Roy. Statist. Soc. Ser. B'' , '''32''' : 1 (1970) pp. 149–152</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> E. Seneta,   "On the supercritical Galton–Watson process with immigration" ''Math. Biosci.'' , '''7''' (1970) pp. 9–14</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> J.H. Foster,   "A limit theorem for a branching process with state-dependent immigration" ''Ann. of Math. Statist.'' , '''42''' : 5 (1971) pp. 1773–1776</TD></TR></table>
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<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A.M. Zubkov, "Life-like periods of a branching process with immigration" ''Theory Probab. Appl.'' , '''17''' : 1 (1972) pp. 174–183 ''Teor. Veroyatnost. i Primenen.'' , '''17''' : 1 (1972) pp. 179–188</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A.G. Pakes, "Further results on the critical Galton–Watson process with immigration" ''J. Austral. Math. Soc.'' , '''13''' : 3 (1972) pp. 277–290 {{MR|0312585}} {{ZBL|0235.60078}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> J.H. Foster, J.A. Williamson, "Limit theorems for the Galton–Watson process with time-dependent immigration" ''Z. Wahrsch. Verw. Geb.'' , '''20''' (1971) pp. 227–235 {{MR|0305494}} {{ZBL|0219.60069}} {{ZBL|0212.19702}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> E. Seneta, "An explicit limit theorem for the critical Galton–Watson process with immigration" ''J. Roy. Statist. Soc. Ser. B'' , '''32''' : 1 (1970) pp. 149–152 {{MR|0266320}} {{ZBL|0198.52002}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> E. Seneta, "On the supercritical Galton–Watson process with immigration" ''Math. Biosci.'' , '''7''' (1970) pp. 9–14 {{MR|0270460}} {{MR|0268974}} {{ZBL|0209.48804}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> J.H. Foster, "A limit theorem for a branching process with state-dependent immigration" ''Ann. of Math. Statist.'' , '''42''' : 5 (1971) pp. 1773–1776 {{MR|0348854}} {{ZBL|0245.60063}} </TD></TR></table>
  
  

Revision as of 10:30, 27 March 2012

2020 Mathematics Subject Classification: Primary: 60J80 [MSN][ZBL]

A model of a branching process (discrete-time or continuous-time, with one or several types of particles, etc.) in which new particles may appear not only as the products of division of the original particles, but also as a result of immigration from some "external source" . For instance, let

be independent random variables with generating functions

respectively; the branching Galton–Watson process with immigration may be defined by the relations , where is the number of particles and

Here, the variable is interpreted as the number of daughter particles of the -th particle of the -th generation, while the variable is interpreted as the number of the particles which have immigrated into the -th generation. The generating functions

are given by the recurrence relations

The Markov chain corresponding to the Galton–Watson branching process with immigration is recurrent if and or and , and is transient if and or . For the Markov chain to be ergodic, i.e. for the limits

to exist and to satisfy

it is necessary and sufficient [3] that

This condition is met, in particular, if and . If , , , then [4]

If and , then there exists [5] a sequence of numbers , , such that

In branching processes with immigration in which the immigration takes place at only, i.e.

where is the Kronecker symbol, the following relation is valid if , and :

References

[1] A.M. Zubkov, "Life-like periods of a branching process with immigration" Theory Probab. Appl. , 17 : 1 (1972) pp. 174–183 Teor. Veroyatnost. i Primenen. , 17 : 1 (1972) pp. 179–188
[2] A.G. Pakes, "Further results on the critical Galton–Watson process with immigration" J. Austral. Math. Soc. , 13 : 3 (1972) pp. 277–290 MR0312585 Zbl 0235.60078
[3] J.H. Foster, J.A. Williamson, "Limit theorems for the Galton–Watson process with time-dependent immigration" Z. Wahrsch. Verw. Geb. , 20 (1971) pp. 227–235 MR0305494 Zbl 0219.60069 Zbl 0212.19702
[4] E. Seneta, "An explicit limit theorem for the critical Galton–Watson process with immigration" J. Roy. Statist. Soc. Ser. B , 32 : 1 (1970) pp. 149–152 MR0266320 Zbl 0198.52002
[5] E. Seneta, "On the supercritical Galton–Watson process with immigration" Math. Biosci. , 7 (1970) pp. 9–14 MR0270460 MR0268974 Zbl 0209.48804
[6] J.H. Foster, "A limit theorem for a branching process with state-dependent immigration" Ann. of Math. Statist. , 42 : 5 (1971) pp. 1773–1776 MR0348854 Zbl 0245.60063


Comments

Additional references may be found in the article Branching process.

How to Cite This Entry:
Branching process with immigration. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Branching_process_with_immigration&oldid=23588
This article was adapted from an original article by A.M. Zubkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article