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Limit Theorems for Weakly Subcritical Branching Processes in Random Environment

Author

Listed:
  • V. I. Afanasyev

    (Steklov Mathematical Institute)

  • C. Böinghoff

    (Universität Frankfurt)

  • G. Kersting

    (Universität Frankfurt)

  • V. A. Vatutin

    (Steklov Mathematical Institute)

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.

Suggested Citation

  • V. I. Afanasyev & C. Böinghoff & G. Kersting & V. A. Vatutin, 2012. "Limit Theorems for Weakly Subcritical Branching Processes in Random Environment," Journal of Theoretical Probability, Springer, vol. 25(3), pages 703-732, September.
    • Handle: RePEc:spr:jotpro:v:25:y:2012:i:3:d:10.1007_s10959-010-0331-6
    DOI: 10.1007/s10959-010-0331-6
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    References listed on IDEAS

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    1. Afanasyev, V.I. & Geiger, J. & Kersting, G. & Vatutin, V.A., 2005. "Functional limit theorems for strongly subcritical branching processes in random environment," Stochastic Processes and their Applications, Elsevier, vol. 115(10), pages 1658-1676, October.
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    Cited by:

    1. Doudou Li & Vladimir Vatutin & Mei Zhang, 2021. "Subcritical Branching Processes in Random Environment with Immigration Stopped at Zero," Journal of Theoretical Probability, Springer, vol. 34(2), pages 874-896, June.
    2. Hui He & Zenghu Li & Wei Xu, 2018. "Continuous-State Branching Processes in Lévy Random Environments," Journal of Theoretical Probability, Springer, vol. 31(4), pages 1952-1974, December.
    3. Xu, Wei, 2023. "Asymptotics for exponential functionals of random walks," Stochastic Processes and their Applications, Elsevier, vol. 165(C), pages 1-42.

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