Difference between revisions of "Branching process with immigration"
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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 {{MR|0300351}} </TD></TR> | + | <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 {{MR|0300351}} {{ZBL|0267.60084}}</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> | <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> | ||
====Comments==== | ====Comments==== | ||
Additional references may be found in the article [[Branching process|Branching process]]. | Additional references may be found in the article [[Branching process|Branching process]]. |
Revision as of 11:58, 29 April 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
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be independent random variables with generating functions
![]() |
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respectively; the branching Galton–Watson process with immigration may be defined by the relations , where
is the number of particles and
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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
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are given by the recurrence relations
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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
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to exist and to satisfy
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it is necessary and sufficient [3] that
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This condition is met, in particular, if and
. If
,
,
, then [4]
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If and
, then there exists [5] a sequence of numbers
,
, such that
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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 MR0300351 Zbl 0267.60084 |
[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.
Branching process with immigration. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Branching_process_with_immigration&oldid=25690