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Moments, moderate and large deviations for a branching process in a random environment

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  • Huang, Chunmao
  • Liu, Quansheng

Abstract

Let (Zn) be a supercritical branching process in a random environment ξ, and W be the limit of the normalized population size Zn/E[Zn|ξ]. We show large and moderate deviation principles for the sequence logZn (with appropriate normalization). For the proof, we calculate the critical value for the existence of harmonic moments of W, and show an equivalence for all the moments of Zn. Central limit theorems on W−Wn and logZn are also established.

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  • 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.
    • Handle: RePEc:eee:spapps:v:122:y:2012:i:2:p:522-545
    DOI: 10.1016/j.spa.2011.09.001
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    References listed on IDEAS

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    1. Liu, Quansheng, 1999. "Asymptotic properties of supercritical age-dependent branching processes and homogeneous branching random walks," Stochastic Processes and their Applications, Elsevier, vol. 82(1), pages 61-87, July.
    2. 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.
    3. Wang, Hesong & Gao, Zhiqiang & Liu, Quansheng, 2011. "Central limit theorems for a supercritical branching process in a random environment," Statistics & Probability Letters, Elsevier, vol. 81(5), pages 539-547, May.
    4. Liu, Quansheng, 2001. "Asymptotic properties and absolute continuity of laws stable by random weighted mean," Stochastic Processes and their Applications, Elsevier, vol. 95(1), pages 83-107, September.
    5. 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.
    6. 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. Li, Yingqiu & Liu, Quansheng & Peng, Xuelian, 2019. "Harmonic moments, large and moderate deviation principles for Mandelbrot’s cascade in a random environment," Statistics & Probability Letters, Elsevier, vol. 147(C), pages 57-65.
    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.
    4. 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.
    5. Peter Eichelsbacher & Matthias Löwe, 2019. "Lindeberg’s Method for Moderate Deviations and Random Summation," Journal of Theoretical Probability, Springer, vol. 32(2), pages 872-897, June.
    6. Gao, Zhenlong & Wang, Weigang, 2015. "Large deviations for a Poisson random indexed branching process," Statistics & Probability Letters, Elsevier, vol. 105(C), pages 143-148.
    7. Struleva, M.A. & Prokopenko, E.I., 2022. "Integro-local limit theorems for supercritical branching process in a random environment," Statistics & Probability Letters, Elsevier, vol. 181(C).
    8. Xiao, Hui & Grama, Ion & Liu, Quansheng, 2021. "Berry–Esseen bounds and moderate deviations for random walks on GLd(R)," Stochastic Processes and their Applications, Elsevier, vol. 142(C), pages 293-318.
    9. Gao, Zhenlong & Zhang, Yanhua, 2015. "Large and moderate deviations for a class of renewal random indexed branching process," Statistics & Probability Letters, Elsevier, vol. 103(C), pages 1-5.
    10. Doukhan, Paul & Fan, Xiequan & Gao, Zhi-Qiang, 2023. "Cramér moderate deviations for a supercritical Galton–Watson process," Statistics & Probability Letters, Elsevier, vol. 192(C).

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