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Teor. Veroyatnost. i Primenen., 1982, Volume 27, Issue 2, Pages 228–238 (Mi tvp2340)  

This article is cited in 4 scientific papers (total in 4 papers)

The limit behaviour of decomposable critical branching processes with two types of particles

A. M. Zubkov

Moscow

Abstract: We consider two-dimensional branching processes $\mu(t)=(\mu_1(t),\mu_2(t))$, $t\in\{0,1,…\}$, with the offspring generating functions
\begin{gather*} \mathbf E\{s_1^{\mu_1(1)}s_2^{\mu_2(1)}\mid\mu(0)=(1,0)\}= F_1(s_1)=s_1+(1-s_1)^{1+\alpha_1}L_1(1-s_1),
\mathbf E\{s_1^{\mu_1(1)}s_2^{\mu_2(1)}\mid\mu(0)=(0,1)\}= s_2+(1-s_2)^{1+\alpha_2}L_2(1-s_2)-(A+o(1))(1-s_1), \end{gather*}
where $0<\alpha_1$, $\alpha_2\le 1$ and the functions $L_1(x)$, $L_2(x)$ are slowly varying when $x\downarrow 0$. We investigate the asymptotics of
$$ \mathbf P\{\mu(t)\ne 0\mid\mu(0)=(0,1)\},\qquad t\to\infty, $$
and prove the limit theorems for the conditional distribution of the numbers of particles.

Full text: PDF file (592 kB)

English version:
Theory of Probability and its Applications, 1983, 27:2, 235–237

Bibliographic databases:

Document Type: Article
Received: 11.05.1981

Citation: A. M. Zubkov, “The limit behaviour of decomposable critical branching processes with two types of particles”, Teor. Veroyatnost. i Primenen., 27:2 (1982), 228–238; Theory Probab. Appl., 27:2 (1983), 235–237

Citation in format AMSBIB
\Bibitem{Zub82}
\by A.~M.~Zubkov
\paper The limit behaviour of decomposable critical branching processes with two types of particles
\jour Teor. Veroyatnost. i Primenen.
\yr 1982
\vol 27
\issue 2
\pages 228--238
\mathnet{http://mi.mathnet.ru/tvp2340}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=657917}
\zmath{https://zbmath.org/?q=an:0502.60067}
\transl
\jour Theory Probab. Appl.
\yr 1983
\vol 27
\issue 2
\pages 235--237
\crossref{https://doi.org/10.1137/1127026}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1983QN71900002}


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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. V. A. Vatutin, E. E. D'yakonova, “Decomposable branching processes with a fixed extinction moment”, Proc. Steklov Inst. Math., 290:1 (2015), 103–124  mathnet  crossref  crossref  isi  elib  elib
    2. Vladimir A. Vatutin, Elena E. Dyakonova, “Extinction of decomposable branching processes”, Discrete Math. Appl., 26:3 (2016), 183–192  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    3. Smadi C. Vatutin V.A., “Reduced Two-Type Decomposable Critical Branching Processes With Possibly Infinite Variance”, Markov Process. Relat. Fields, 22:2 (2016), 311–358  mathscinet  zmath  isi
    4. V. A. Vatutin, “A Conditional Functional Limit Theorem for Decomposable Branching Processes with Two Types of Particles”, Math. Notes, 101:5 (2017), 778–789  mathnet  crossref  crossref  mathscinet  isi  elib
  • Теория вероятностей и ее применения Theory of Probability and its Applications
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