# 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

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 |

[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 |

[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 |

[5] | E. Seneta, "On the supercritical Galton–Watson process with immigration" Math. Biosci. , 7 (1970) pp. 9–14 |

[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 |

#### 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=13912