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Teoriya Veroyatnostei i ee Primeneniya, 2008, Volume 53, Issue 4, Pages 665–683
DOI: https://doi.org/10.4213/tvp2459
(Mi tvp2459)
 

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

Waves in Reduced Branching Processes in a Random Environment

V. A. Vatutin, E. E. D'yakonova

Steklov Mathematical Institute, Russian Academy of Sciences
References:
Abstract: Let $Z(n)$, $n=0,1\dots,$ be a branching process evolving in the random environment generated by a sequence of independent identically distributed generating functions $f_{0}(s),f_{1}(s),\dots,$ and let $S_{0}=0$, $S_{k}=X_{1}+\dots+X_{k}$, $k\ge1,$ be the associated random walk with $X_{i}=\log f_{i-1}'(1),$ and $\tau (n)$ be the leftmost point of the minimum of $\{ S_{k}$,$k\ge0\} $ on the interval $[0,n]$. Denoting by $Z(k,m)$ the number of particles existing in the branching process at the time moment $k\le m$ which have nonempty offspring at the time moment $m$, and assuming that the associated random walk satisfies the Doney condition $P(S_{n}>0)\to \rho \in (0,1)$, $n\to\infty$, we prove (under the quenched approach) conditional limit theorems, as $n\to\infty$, for the distribution of $Z(nt_{1},nt_{2})$, $0<t_{1}<t_{2}<1,$ given $Z(n)>0$. It is shown that the form of the limit distributions essentially depends on the position of $\tau (n)$ with respect to the interval $[nt_{1},nt_{2}].$
Keywords: branching processes in a random environment, Doney condition, conditional limit theorems.
Received: 23.04.2007
English version:
Theory of Probability and its Applications, 2009, Volume 53, Issue 4, Pages 679–695
DOI: https://doi.org/10.1137/S0040585X97983845
Bibliographic databases:
Document Type: Article
Language: Russian
Citation: V. A. Vatutin, E. E. D'yakonova, “Waves in Reduced Branching Processes in a Random Environment”, Teor. Veroyatnost. i Primenen., 53:4 (2008), 665–683; Theory Probab. Appl., 53:4 (2009), 679–695
Citation in format AMSBIB
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\pages 665--683
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\jour Theory Probab. Appl.
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  • This publication is cited in the following 4 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Теория вероятностей и ее применения Theory of Probability and its Applications
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