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 Teor. Veroyatnost. i Primenen.: Year: Volume: Issue: Page: Find

 Teor. Veroyatnost. i Primenen., 1980, Volume 25, Issue 3, Pages 523–534 (Mi tvp1092)

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 $A_0\in(0,\infty)$
\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

Bibliographic databases:

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
\Bibitem{NagKha80} \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 \mathnet{http://mi.mathnet.ru/tvp1092} \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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Erratum

This publication is cited in the following articles:
1. Yanev G.P., Yanev N.M., “A critical branching process with stationary–limiting distribution”, Stochastic Analysis and Applications, 22:3 (2004), 721–738
2. 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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