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Functional limit theorems for strongly subcritical branching processes in random environment

Author

Listed:
  • Afanasyev, V.I.
  • Geiger, J.
  • Kersting, G.
  • Vatutin, V.A.

Abstract

For a strongly subcritical branching process (Zn)n[greater-or-equal, slanted]0 in random environment the non-extinction probability at generation n decays at the same exponential rate as the expected generation size and given non-extinction at n the conditional distribution of Zn has a weak limit. Here we prove conditional functional limit theorems for the generation size process (Zk)0[less-than-or-equals, slant]k[less-than-or-equals, slant]n as well as for the random environment. We show that given the population survives up to generation n the environmental sequence still evolves in an i.i.d. fashion and that the conditioned generation size process converges in distribution to a positive recurrent Markov chain.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:115:y:2005:i:10:p:1658-1676
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    Citations

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    Cited by:

    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. Böinghoff, Christian, 2014. "Limit theorems for strongly and intermediately supercritical branching processes in random environment with linear fractional offspring distributions," Stochastic Processes and their Applications, Elsevier, vol. 124(11), pages 3553-3577.
    3. Li, Zenghu & Xu, Wei, 2018. "Asymptotic results for exponential functionals of Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 128(1), pages 108-131.
    4. Alsmeyer, Gerold & Gröttrup, Sören, 2016. "Branching within branching: A model for host–parasite co-evolution," Stochastic Processes and their Applications, Elsevier, vol. 126(6), pages 1839-1883.
    5. Xu, Wei, 2023. "Asymptotics for exponential functionals of random walks," Stochastic Processes and their Applications, Elsevier, vol. 165(C), pages 1-42.
    6. Wang, Yuejiao & Liu, Zaiming & Li, Yingqiu & Liu, Quansheng, 2017. "On the concept of subcriticality and criticality and a ratio theorem for a branching process in a random environment," Statistics & Probability Letters, Elsevier, vol. 127(C), pages 97-103.
    7. Huang, Chunmao & Liu, Quansheng, 2012. "Moments, moderate and large deviations for a branching process in a random environment," Stochastic Processes and their Applications, Elsevier, vol. 122(2), pages 522-545.
    8. Böinghoff, Christian & Kersting, Götz, 2010. "Upper large deviations of branching processes in a random environment--Offspring distributions with geometrically bounded tails," Stochastic Processes and their Applications, Elsevier, vol. 120(10), pages 2064-2077, September.
    9. 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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