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Teor. Veroyatnost. i Primenen., 1983, Volume 28, Issue 2, Pages 382–388 (Mi tvp2304)  

This article is cited in 2 scientific papers (total in 2 papers)

Short Communications

Limit theorems for a sequence of branching processes with immigration

I. S. Badalbaeva, A. M. Zubkovb

a Taškent
b Moscow

Abstract: We consider a family $Z^{(n)}( \cdot )$ of branching processes with immigration defined by a formula
$$ Z^{(n)}(t)=\sum_{k\colon\theta_k^{(n)}\le t}\zeta_k^{(n)}(t-\theta_k^{(n)}), $$
where $\theta_k^{(n)}$ – the moment of immigration of k$^{th}$ particle and $\zeta_k^{(n)}( \cdot )$ – a branching process of its descendants. It is supposed that:
$$ i)\quad \mathbf P\{0\le\theta_1^{(n)}\le\theta_2^{(n)}\le\dotsb, \lim_{k\to\infty}\theta_k^{(n)}\}=1 $$
and all finite-dimensional distributions of the processes
$$ \tau^{(n)}(\alpha)=n^{-1}\sum_{k\colon\theta_k^{(n)}\le\alpha n}1 $$
converge to the corresponding finite-dimensional distrutions of a random process $T(\alpha)$, $\alpha\in[0,1]$ which is stochastically continuous at $\alpha=1$;
$$ ii)\quad \mathbf Ms^{\xi_k^{(n)}(t)}=1-\frac{1-s}{1+(1-s)t\gamma}(1+\alpha_n(t;s)), $$
where $\gamma=\mathrm{const}$ and $\alpha_n(t;s)\to 0$, $n\to\infty$, uniformly in the set $\{\varepsilon n\le t\le n, |s|\le 1\}$ for every $\varepsilon>0$.
Theorem 1. If the conditions i) and ii) are fulfilled, then
$$ \lim_{n\to\infty}\mathbf M\exp\{-u\frac{Z^{(n)}(n)}{n\gamma}\}=\mathbf M\exp\{-\frac{u}{\gamma}\int_0^1\frac{dT(s)}{1+(1-s)u}\}. $$
Some generalizations are considered also.

Full text: PDF file (1084 kB)

English version:
Theory of Probability and its Applications, 1984, 28:2, 404–409

Bibliographic databases:

Received: 27.04.1982

Citation: I. S. Badalbaev, A. M. Zubkov, “Limit theorems for a sequence of branching processes with immigration”, Teor. Veroyatnost. i Primenen., 28:2 (1983), 382–388; Theory Probab. Appl., 28:2 (1984), 404–409

Citation in format AMSBIB
\Bibitem{BadZub83}
\by I.~S.~Badalbaev, A.~M.~Zubkov
\paper Limit theorems for a sequence of branching processes with immigration
\jour Teor. Veroyatnost. i Primenen.
\yr 1983
\vol 28
\issue 2
\pages 382--388
\mathnet{http://mi.mathnet.ru/tvp2304}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=700219}
\zmath{https://zbmath.org/?q=an:0533.60092|0511.60076}
\transl
\jour Theory Probab. Appl.
\yr 1984
\vol 28
\issue 2
\pages 404--409
\crossref{https://doi.org/10.1137/1128034}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1984SS85900014}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. O. A. Butkovskii, “Limit Behavior of a Critical Branching Process with Immigration”, Math. Notes, 92:5 (2012), 612–618  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    2. Ya. M. Khusanbaev, “On asymptotics of branching processes with immigration”, Discrete Math. Appl., 27:2 (2017), 73–80  mathnet  crossref  crossref  mathscinet  isi  elib
  • Теория вероятностей и ее применения Theory of Probability and its Applications
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