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Limit theorems for a critical branching process with immigration at zero in a random environment

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  • Zhao, Yinxuan
  • Zhang, Mei

Abstract

In this paper, we study a critical branching process with immigration at zero in a random environment Z, where the immigration is allowed entering a generation iff the previous generation is empty. Firstly, we analyze the asymptotic behavior of the survival probability at generation n of Z̃, a critical branching process in a random environment with random initial number of particles. Then, the convergence rate of the transition probabilities pk,0(n) (k≥0) of Z are obtained, which can be compared with the corresponding results for non-random case in literatures.

Suggested Citation

  • Zhao, Yinxuan & Zhang, Mei, 2025. "Limit theorems for a critical branching process with immigration at zero in a random environment," Statistics & Probability Letters, Elsevier, vol. 224(C).
    • Handle: RePEc:eee:stapro:v:224:y:2025:i:c:s016771522500080x
    DOI: 10.1016/j.spl.2025.110435
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    References listed on IDEAS

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    1. 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.
    2. Papastavrou, Jason D., 1996. "A stochastic and dynamic routing policy using branching processes with state dependent immigration," European Journal of Operational Research, Elsevier, vol. 95(1), pages 167-177, November.
    3. 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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