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This article is cited in 2 scientific papers (total in 2 papers)
Limit theorems for a critical Galton–Watson process with migration
S. V. Nagaev, L. V. Han Novosibirsk
Abstract:
The critical Galton–Watson process with immigration and emigration is investigated.
We consider the population of particles which develop according to the critical
Galton–Watson process with the offspring generating function $f(s)$, and at each moment
$n=0,1,…$ either $k$ ($k=0,1,…$) particles immigrate in the population with the
probability $p_k$ or $j$ ($j=1,…,m$) particles of those present at time $n$ emigrate from the
population with probability $q_j$, where $m$ is a fixed natural number,
$$
\sum_{k=0}^\infty p_k+\sum_{k=1}^m q_k=1,\qquad q_m>0.
$$
Let $Z_n$ ($n=0,1,…$) be the number of particles at time $n$. We suppose that
$$
Z_0=0,\qquad f'(1-)=1,\qquad\sum_{k=1}^\infty kp_k-\sum_{k=1}^m kq_k=0.
$$
The following results are obtained. If
$$
f(0)>0,\qquad B=1/2f"(1-)<\infty,\qquad\sum_{k=1}^\infty k^2p_k<\infty,
$$
then for some $A_0\in(0,\infty)$
\begin{gather*}
\mathbf PŻ_n=0\}\sim\frac{A_0}{\log n},\quad\mathbf MZ_n\sim\frac{B_n}{\log n},\quad\mathbf DZ_n\sim\frac{2B^2n^2}{\log n}\quad(n\to\infty),
\lim_{n\to\infty}\mathbf P\{\frac{\log Z_n}{\log n}<x\}=x,\qquad x\in[0,1].
\end{gather*}
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English version:
Theory of Probability and its Applications, 1981, 25:3, 514–525
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Citation:
S. V. Nagaev, L. V. Han, “Limit theorems for a critical Galton–Watson process with migration”, Teor. Veroyatnost. i Primenen., 25:3 (1980), 523–534; Theory Probab. Appl., 25:3 (1981), 514–525
Citation in format AMSBIB
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\by S.~V.~Nagaev, L.~V.~Han
\paper Limit theorems for a~critical Galton--Watson process with migration
\jour Teor. Veroyatnost. i Primenen.
\yr 1980
\vol 25
\issue 3
\pages 523--534
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\mathscinet{http://www.ams.org/mathscinet-getitem?mr=582582}
\zmath{https://zbmath.org/?q=an:0462.60082|0436.60059}
\transl
\jour Theory Probab. Appl.
\yr 1981
\vol 25
\issue 3
\pages 514--525
\crossref{https://doi.org/10.1137/1125063}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1980MB70100007}
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http://mi.mathnet.ru/eng/tvp1092 http://mi.mathnet.ru/eng/tvp/v25/i3/p523
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Erratum
This publication is cited in the following articles:
-
Yanev G.P., Yanev N.M., “A critical branching process with stationary–limiting distribution”, Stochastic Analysis and Applications, 22:3 (2004), 721–738
-
Nagaev S.V., “On probability and moment inequalities for supermartingales and martingales”, Acta Applicandae Mathematicae, 97:1–3 (2007), 151–162
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