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Berry–Esseen’s bound and Cramér’s large deviation expansion for a supercritical branching process in a random environment

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  • Grama, Ion
  • Liu, Quansheng
  • Miqueu, Eric

Abstract

Let (Zn) be a supercritical branching process in a random environment ξ=(ξn). We establish a Berry–Esseen bound and a Cramér’s type large deviation expansion for logZn under the annealed law P. We also improve some earlier results about the harmonic moments of the limit variable W=limn→∞Wn, where Wn=Zn/EξZn is the normalized population size.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:127:y:2017:i:4:p:1255-1281
    DOI: 10.1016/j.spa.2016.07.014
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    References listed on IDEAS

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    1. Tanny, David, 1988. "A necessary and sufficient condition for a branching process in a random environment to grow like the product of its means," Stochastic Processes and their Applications, Elsevier, vol. 28(1), pages 123-139, April.
    2. Nakashima, Makoto, 2013. "Lower deviations of branching processes in random environment with geometrical offspring distributions," Stochastic Processes and their Applications, Elsevier, vol. 123(9), pages 3560-3587.
    3. 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.
    4. 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.
    5. 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.
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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. Doukhan, Paul & Fan, Xiequan & Gao, Zhi-Qiang, 2023. "Cramér moderate deviations for a supercritical Galton–Watson process," Statistics & Probability Letters, Elsevier, vol. 192(C).
    3. 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.

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