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Teor. Veroyatnost. i Primenen., 2008, Volume 53, Issue 4, Pages 665–683 (Mi tvp2459)  

This article is cited in 3 scientific papers (total in 3 papers)

Waves in Reduced Branching Processes in a Random Environment

V. A. Vatutin, E. E. D'yakonova

Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: Let $Z(n)$, $n=0,1…,$ be a branching process evolving in the random environment generated by a sequence of independent identically distributed generating functions $f_{0}(s),f_{1}(s),…,$ and let $S_{0}=0$, $S_{k}=X_{1}+…+X_{k}$, $k\ge1,$ 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.

DOI: https://doi.org/10.4213/tvp2459

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English version:
Theory of Probability and its Applications, 2009, 53:4, 679–695

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Received: 23.04.2007

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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\by V.~A.~Vatutin, E.~E.~D'yakonova
\paper Waves in Reduced Branching Processes in a Random Environment
\jour Teor. Veroyatnost. i Primenen.
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\issue 4
\pages 665--683
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\transl
\jour Theory Probab. Appl.
\yr 2009
\vol 53
\issue 4
\pages 679--695
\crossref{https://doi.org/10.1137/S0040585X97983845}
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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. Dyakonova, S. Sagitov, “Evolution of branching processes in a random environment”, Proc. Steklov Inst. Math., 282 (2013), 220–242  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    2. Elena E. D'yakonova, “Reduced multitype critical branching processes in random environment”, Discrete Math. Appl., 28:1 (2018), 7–22  mathnet  crossref  crossref  mathscinet  isi  elib
    3. V. A. Vatutin, E. E. D'yakonova, “How many families survive for a long time?”, Theory Probab. Appl., 61:4 (2017), 692–711  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
  • Теория вероятностей и ее применения Theory of Probability and its Applications
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