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Exact convergence rate in the central limit theorem for a branching process in a random environment

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  • Gao, Zhi-Qiang

Abstract

Let {Zn} be a supercritical branching process in an independent and identically distributed random environment. As is well known, the behavior of Zn depends primarily on that of the associated random walk Sn constructed by the logarithms of the quenched expectation of population sizes. By this observation, the Berry–Esséen bound for logZn has been established by Grama et al. (2017). To refine that, we figure out the exact convergence rate in the central limit theorem for logZn under the annealed law, with less restrictive moment conditions. In particular, there is one factor in the rate function concerning on logZn that does not appear in that for Sn. Hence the result indicates the essential difference between logZn and Sn.

Suggested Citation

  • 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).
    • Handle: RePEc:eee:stapro:v:178:y:2021:i:c:s0167715221001565
    DOI: 10.1016/j.spl.2021.109194
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    References listed on IDEAS

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    1. 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.
    2. 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.
    3. 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.
    4. 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.
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