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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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[[Category:Branching processes]]
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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
 +
 
 +
$$
 +
X _ {t,i }  , Y _ {t} ,\  t = 0, 1 ,\dots ; \  i = 1, 2 \dots
 +
$$
  
 
be independent random variables with generating functions
 
be independent random variables with generating functions
  
<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/b0175802.png" /></td> </tr></table>
+
$$
 +
F (s)  = \
 +
\sum _ {k = 0 } ^  \infty 
 +
{\mathsf P} \{ X _ {t,i }  = k \} s  ^ {k} ,
 +
$$
  
<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/b0175803.png" /></td> </tr></table>
+
$$
 +
G (s)  = \sum _ {k = 0 } ^  \infty  {\mathsf P} \{ Y _ {t} = k \} s  ^ {k} ,
 +
$$
  
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
+
respectively; the branching [[Galton–Watson process|Galton–Watson process]] with immigration may be defined by the relations $  \mu (0) = 0 $,  
 +
where $  \mu (t) $
 +
is the number of particles and
  
<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/b0175806.png" /></td> </tr></table>
+
$$
 +
\mu (t + 1)  = \
 +
X _ {t,1 }  + \dots +
 +
X _ {t, \mu (t) }  + Y _ {t} ,\ \
 +
t=0, 1 ,\dots .
 +
$$
  
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
+
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
  
<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/b01758012.png" /></td> </tr></table>
+
$$
 +
H _ {t} (s)  = \
 +
{\mathsf E} \{ s ^ {\mu (t) }
 +
\mid  \mu (0) = 0 \}
 +
$$
  
 
are given by the recurrence relations
 
are given by the recurrence relations
  
<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/b01758013.png" /></td> </tr></table>
+
$$
 +
H _ {0} (s)  = 1,\ \
 +
H _ {t + 1 }  (s)  = \
 +
G (s) H _ {t} (F (s)).
 +
$$
  
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
+
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
  
<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/b01758023.png" /></td> </tr></table>
+
$$
 +
\lim\limits _ {t \rightarrow \infty } \
 +
{\mathsf P} \{ \mu (t) = k \}  = p _ {k}  $$
  
 
to exist and to satisfy
 
to exist and to satisfy
  
<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/b01758024.png" /></td> </tr></table>
+
$$
 +
\sum _ {k = 0 } ^  \infty 
 +
p _ {k}  = 1,
 +
$$
  
it is necessary and sufficient [[#References|[3]]] that
+
it is necessary and sufficient {{Cite|FW}} that
  
<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/b01758025.png" /></td> </tr></table>
+
$$
 +
\int\limits _ { 0 } ^ { 1 }
  
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]]]
+
\frac{1 - G (s) }{F (s) - s }
 +
\
 +
ds  < \infty .
 +
$$
  
<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/b01758031.png" /></td> </tr></table>
+
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 {{Cite|S}}
  
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
+
$$
 +
\lim\limits _ {t \rightarrow \infty }  {\mathsf P}
 +
\left \{
 +
\frac{2 \mu (t) }{Bt }
 +
\leq  x \right \}  = \
 +
{
 +
\frac{1}{\Gamma (2CB  ^ {-1} ) }
 +
}
 +
\int\limits _ { 0 } ^ { x }
 +
y ^ {2CB ^ {-1 } -1 }
 +
e  ^ {-y}  dy,\  x \geq  0.
 +
$$
  
<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/b01758036.png" /></td> </tr></table>
+
If  $  A = {\mathsf E} X _ {t,i} > 1 $
 +
and  $  {\mathsf E}  \mathop{\rm ln} (1 + Y _ {t} ) < \infty $,
 +
then there exists {{Cite|S2}} a sequence of numbers  $  c _ {t} \downarrow 0 $,
 +
$  c _ {t} / c _ {t+1} \rightarrow A $,
 +
such that
  
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.
+
$$
 +
{\mathsf P} \left \{
 +
\lim\limits _ {t \rightarrow \infty } \
 +
c _ {t} \mu (t) \
 +
\textrm{ exists }  \textrm{ and } \
 +
\textrm{ is }  \textrm{ positive }
 +
\right \}  = 1.
 +
$$
  
<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/b01758038.png" /></td> </tr></table>
+
In branching processes with immigration in which the immigration takes place at  $  \mu (t) = 0 $
 +
only, i.e.
  
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" />:
+
$$
 +
\mu (t+1)  = X _ {t,1} + \dots + X _ {t, \mu (t) }  + \delta _ {0, \mu (t) }
 +
Y _ {t} ,\  t=0, 1 \dots
 +
$$
  
<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/b01758043.png" /></td> </tr></table>
+
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 $:
  
====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>
+
\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====
 +
{|
 +
|valign="top"|{{Ref|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 {{MR|0300351}} {{ZBL|0267.60084}}
 +
|-
 +
|valign="top"|{{Ref|P}}|| 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}}
 +
|-
 +
|valign="top"|{{Ref|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 {{MR|0305494}} {{ZBL|0219.60069}} {{ZBL|0212.19702}}
 +
|-
 +
|valign="top"|{{Ref|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 {{MR|0266320}} {{ZBL|0198.52002}}
 +
|-
 +
|valign="top"|{{Ref|S2}}|| E. Seneta, "On the supercritical Galton–Watson process with immigration" ''Math. Biosci.'' , '''7''' (1970) pp. 9–14 {{MR|0270460}} {{ZBL|0209.48804}}
 +
|-
 +
|valign="top"|{{Ref|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 {{MR|0348854}} {{ZBL|0245.60063}}
 +
|}
  
 
====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]].

Latest revision as of 06:29, 30 May 2020


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.

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
This article was adapted from an original article by A.M. Zubkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article