Branching process with immigration

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

$$ X _ {t,i } , Y _ {t} ,\ t = 0, 1 ,\dots ; \ i = 1, 2 \dots $$

be independent random variables with generating functions

$$ F (s) = \ \sum _ {k = 0 } ^ \infty {\mathsf P} \{ X _ {t,i } = k \} s ^ {k} , $$

$$ G (s) = \sum _ {k = 0 } ^ \infty {\mathsf P} \{ Y _ {t} = k \} s ^ {k} , $$

respectively; the branching Galton–Watson process with immigration may be defined by the relations $ \mu (0) = 0 $, where $ \mu (t) $ is the number of particles and

$$ \mu (t + 1) = \ X _ {t,1 } + \dots + X _ {t, \mu (t) } + Y _ {t} ,\ \ t=0, 1 ,\dots . $$

Here, the variable $ X _ {t,i} $ is interpreted as the number of daughter particles of the $ i $- th particle of the $ t $- th generation, while the variable $ Y _ {t} $ is interpreted as the number of the particles which have immigrated into the $ t $- th generation. The generating functions

$$ H _ {t} (s) = \ {\mathsf E} \{ s ^ {\mu (t) } \mid \mu (0) = 0 \} $$

are given by the recurrence relations

$$ H _ {0} (s) = 1,\ \ H _ {t + 1 } (s) = \ G (s) H _ {t} (F (s)). $$

The Markov chain $ \mu (t) $ corresponding to the Galton–Watson branching process with immigration is recurrent if $ {\mathsf E} X _ {t,i} < 1 $ and $ {\mathsf E} \mathop{\rm ln} (1 + Y _ {t} ) < \infty $ or $ {\mathsf E} X _ {t,i} = 1 $ and $ B = {\mathsf D} X _ {t,i} > 2C = 2 {\mathsf E} Y _ {t} $, and is transient if $ {\mathsf E} X _ {t,i} = 1 $ and $ B < 2C $ or $ {\mathsf E} X _ {t,i} > 1 $. For the Markov chain $ \mu (t) $ to be ergodic, i.e. for the limits

$$ \lim\limits _ {t \rightarrow \infty } \ {\mathsf P} \{ \mu (t) = k \} = p _ {k} $$

to exist and to satisfy

$$ \sum _ {k = 0 } ^ \infty p _ {k} = 1, $$

it is necessary and sufficient [FW] that

$$ \int\limits _ { 0 } ^ { 1 } \frac{1 - G (s) }{F (s) - s } \ ds < \infty . $$

This condition is met, in particular, if $ {\mathsf E} X _ {t,i} < 1 $ and $ {\mathsf E} \mathop{\rm ln} (1 + Y _ {t} ) < \infty $. If $ {\mathsf E} X _ {t,i} = 1 $, $ B > 0 $, $ C < \infty $, then [S]

$$ \lim\limits _ {t \rightarrow \infty } {\mathsf P} \left \{ \frac{2 \mu (t) }{Bt } \leq x \right \} = \ { \frac{1}{\Gamma (2CB ^ {-1} ) } } \int\limits _ { 0 } ^ { x } y ^ {2CB ^ {-1 } -1 } e ^ {-y} dy,\ x \geq 0. $$

If $ A = {\mathsf E} X _ {t,i} > 1 $ and $ {\mathsf E} \mathop{\rm ln} (1 + Y _ {t} ) < \infty $, then there exists [S2] a sequence of numbers $ c _ {t} \downarrow 0 $, $ c _ {t} / c _ {t+1} \rightarrow A $, such that

$$ {\mathsf P} \left \{ \lim\limits _ {t \rightarrow \infty } \ c _ {t} \mu (t) \ \textrm{ exists } \textrm{ and } \ \textrm{ is } \textrm{ positive } \right \} = 1. $$

In branching processes with immigration in which the immigration takes place at $ \mu (t) = 0 $ only, i.e.

$$ \mu (t+1) = X _ {t,1} + \dots + X _ {t, \mu (t) } + \delta _ {0, \mu (t) } Y _ {t} ,\ t=0, 1 \dots $$

where $ \delta _ {ij} $ is the Kronecker symbol, the following relation is valid if $ {\mathsf E} X _ {t,i} = 1 $, $ 1 < {\mathsf E} X _ {t,i} ^ { 2 } < \infty $ and $ 0 < {\mathsf E} Y _ {t} < \infty $:

$$ \lim\limits _ {t \rightarrow \infty } \ {\mathsf P} \left \{ \frac{ \mathop{\rm ln} (1 + \mu (t)) }{ \mathop{\rm ln} t } \leq x \right \} = x,\ 0 \leq x \leq 1. $$

References

[Z]	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
[P]	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
[FW]	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
[S]	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
[S2]	E. Seneta, "On the supercritical Galton–Watson process with immigration" Math. Biosci. , 7 (1970) pp. 9–14 MR0270460 Zbl 0209.48804
[F]	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.