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Limit Theorems for Weakly Subcritical Branching Processes in Random Environment

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Abstract

For a branching process in random environment, it is assumed that the offspring distribution of the individuals varies in a random fashion, independently from one generation to the other. Interestingly there is the possibility that the process may at the same time be subcritical and, conditioned on nonextinction, “supercritical.” This so-called weakly subcritical case is considered in this paper. We study the asymptotic survival probability and the size of the population conditioned on nonextinction. Also a functional limit theorem is proved, which makes the conditional supercriticality manifest. A main tool is a new type of functional limit theorems for conditional random walks.

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Authors and Affiliations

  1. Department of Discrete Mathematics, Steklov Mathematical Institute, 8 Gubkin Street, 119991, Moscow, Russia

    V. I. Afanasyev & V. A. Vatutin

  2. Fachbereich Mathematik, Universität Frankfurt, Fach 187, 60054, Frankfurt am Main, Germany

    C. Böinghoff & G. Kersting

Authors
  1. V. I. Afanasyev
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  2. C. Böinghoff
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  3. G. Kersting
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  4. V. A. Vatutin
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Corresponding author

Correspondence to C. Böinghoff.

Additional information

This paper is part of a project supported by the German Research Foundation (DFG) and the Russian Foundation of Basic Research (Grant DFG-RFBR 08-01-91954).

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Afanasyev, V.I., Böinghoff, C., Kersting, G. et al. Limit Theorems for Weakly Subcritical Branching Processes in Random Environment. J Theor Probab 25, 703–732 (2012). https://doi.org/10.1007/s10959-010-0331-6

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  • DOI: https://doi.org/10.1007/s10959-010-0331-6

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