View source for Branching process with immigration

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|Galton–Watson process]] with immigration may be defined by the relations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017580/b0175804.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017580/b0175805.png" /> is the number of particles and

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Here, the variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017580/b0175807.png" /> is interpreted as the number of daughter particles of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017580/b0175808.png" />-th particle of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017580/b0175809.png" />-th generation, while the variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017580/b01758010.png" /> is interpreted as the number of the particles which have immigrated into the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017580/b01758011.png" />-th generation. The generating functions

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are given by the recurrence relations

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The Markov chain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017580/b01758014.png" /> corresponding to the Galton–Watson branching process with immigration is recurrent if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017580/b01758015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017580/b01758016.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017580/b01758017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017580/b01758018.png" />, and is transient if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017580/b01758019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017580/b01758020.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017580/b01758021.png" />. For the Markov chain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017580/b01758022.png" /> 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 [[#References|[3]]] that

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This condition is met, in particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017580/b01758026.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017580/b01758027.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017580/b01758028.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017580/b01758029.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017580/b01758030.png" />, then [[#References|[4]]]

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If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017580/b01758032.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017580/b01758033.png" />, then there exists [[#References|[5]]] a sequence of numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017580/b01758034.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017580/b01758035.png" />, such that

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In branching processes with immigration in which the immigration takes place at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017580/b01758037.png" /> only, i.e.

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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017580/b01758039.png" /> is the Kronecker symbol, the following relation is valid if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017580/b01758040.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017580/b01758041.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017580/b01758042.png" />:

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====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>



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