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

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:

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
\Bibitem{VatDya08} \by V.~A.~Vatutin, E.~E.~D'yakonova \paper Waves in Reduced Branching Processes in a Random Environment \jour Teor. Veroyatnost. i Primenen. \yr 2008 \vol 53 \issue 4 \pages 665--683 \mathnet{http://mi.mathnet.ru/tvp2459} \crossref{https://doi.org/10.4213/tvp2459} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2766140} \zmath{https://zbmath.org/?q=an:1191.60125} \transl \jour Theory Probab. Appl. \yr 2009 \vol 53 \issue 4 \pages 679--695 \crossref{https://doi.org/10.1137/S0040585X97983845} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000273141700007} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-73549084265} 

• http://mi.mathnet.ru/eng/tvp2459
• https://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:
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
2. Elena E. D'yakonova, “Reduced multitype critical branching processes in random environment”, Discrete Math. Appl., 28:1 (2018), 7–22
3. 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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