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 Tr. Mat. Inst. Steklova, 2013, Volume 282, Pages 10–21 (Mi tm3487)

High level subcritical branching processes in a random environment

V. I. Afanasyev

Steklov Mathematical Institute of the Russian Academy of Sciences, Moscow, Russia

Abstract: A subcritical branching process in a random environment is considered under the assumption that the moment-generating function of a step of the associated random walk $\Theta(t)$, $t\geq0$, is equal to 1 for some value of the argument $\varkappa>0$. Let $T_x$ be the time when the process first attains the half-axis $(x,+\infty)$ and $T$ be the lifetime of this process. It is shown that the random variable $T_x/\ln x$, considered under the condition $T_x<+\infty$, converges in distribution to a degenerate random variable equal to $1/\Theta'(\varkappa)$, and the random variable $T/\ln x$, considered under the same condition, converges in distribution to a degenerate random variable equal to $1/\Theta'(\varkappa)-1/\Theta'(0)$.

 Funding Agency Grant Number Russian Foundation for Basic Research 11-01-00139 This work was supported by the Russian Foundation for Basic Research, project no. 11-01-00139.

DOI: https://doi.org/10.1134/S0371968513030023

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English version:
Proceedings of the Steklov Institute of Mathematics, 2013, 282, 4–14

Bibliographic databases:

Document Type: Article
UDC: 519.218.27

Citation: V. I. Afanasyev, “High level subcritical branching processes in a random environment”, Branching processes, random walks, and related problems, Collected papers. Dedicated to the memory of Boris Aleksandrovich Sevastyanov, corresponding member of the Russian Academy of Sciences, Tr. Mat. Inst. Steklova, 282, MAIK Nauka/Interperiodica, Moscow, 2013, 10–21; Proc. Steklov Inst. Math., 282 (2013), 4–14

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/tm3487
• https://doi.org/10.1134/S0371968513030023
• http://mi.mathnet.ru/eng/tm/v282/p10

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This publication is cited in the following articles:
1. V. I. Afanasyev, “Functional limit theorems for high-level subcritical branching processes in random environment”, Discrete Math. Appl., 24:5 (2014), 257–272
2. Vatutin V., “Subcritical Branching Processes in Random Environment”, Branching Processes and Their Applications, Lecture Notes in Statistics, 219, eds. DelPuerto I., Gonzalez M., Gutierrez C., Martinez R., Minuesa C., Molina M., Mota M., Ramos A., Springer, 2016, 97–115
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