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Subcritical branching processes in a random environment without the Cramer condition

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  • Vatutin, Vladimir
  • Zheng, Xinghua

Abstract

A subcritical branching process in random environment (BPRE) is considered whose associated random walk does not satisfy the Cramer condition. The asymptotics for the survival probability of the process is investigated, and a Yaglom type conditional limit theorem is proved for the number of particles up to moment n given survival to this moment. Contrary to other types of subcritical BPRE, the limiting distribution is not discrete. We also show that the process survives for a long time owing to a single big jump of the associate random walk accompanied by a population explosion at the beginning of the process.

Suggested Citation

  • Vatutin, Vladimir & Zheng, Xinghua, 2012. "Subcritical branching processes in a random environment without the Cramer condition," Stochastic Processes and their Applications, Elsevier, vol. 122(7), pages 2594-2609.
    • Handle: RePEc:eee:spapps:v:122:y:2012:i:7:p:2594-2609
    DOI: 10.1016/j.spa.2012.04.008
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    References listed on IDEAS

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    1. Bansaye, Vincent, 2009. "Surviving particles for subcritical branching processes in random environment," Stochastic Processes and their Applications, Elsevier, vol. 119(8), pages 2436-2464, August.
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    Cited by:

    1. Gao, Zhi-Qiang, 2021. "Exact convergence rate in the central limit theorem for a branching process in a random environment," Statistics & Probability Letters, Elsevier, vol. 178(C).
    2. 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.
    3. Grama, Ion & Liu, Quansheng & Miqueu, Eric, 2017. "Berry–Esseen’s bound and Cramér’s large deviation expansion for a supercritical branching process in a random environment," Stochastic Processes and their Applications, Elsevier, vol. 127(4), pages 1255-1281.

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