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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
Bibliographic databases:
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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http://mi.mathnet.ru/eng/tvp2459https://doi.org/10.4213/tvp2459 http://mi.mathnet.ru/eng/tvp/v53/i4/p665
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This publication is cited in the following articles:
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V. A. Vatutin, E. E. Dyakonova, S. Sagitov, “Evolution of branching processes in a random environment”, Proc. Steklov Inst. Math., 282 (2013), 220–242
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Elena E. D'yakonova, “Reduced multitype critical branching processes in random environment”, Discrete Math. Appl., 28:1 (2018), 7–22
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V. A. Vatutin, E. E. D'yakonova, “How many families survive for a long time?”, Theory Probab. Appl., 61:4 (2017), 692–711
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