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Uniform Cramér moderate deviations and Berry-Esseen bounds for a supercritical branching process in a random environment

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Abstract

Let {Zn, n ⩾ 0} be a supercritical branching process in an independent and identically distributed random environment. We prove Cramér moderate deviations and Berry-Esseen bounds for log(\(\left( {{Z_{n + {n_0}}}/{Z_{{n_0}}}} \right)\)) uniformly in n0 ∈ ℕ, which extend the corresponding results by I. Grama, Q. Liu, and M. Miqueu [Stochastic Process. Appl., 2017, 127: 1255–1281] established for n0 = 0. The extension is interesting in theory, and is motivated by applications. A new method is developed for the proofs; some conditions of Grama et al. are relaxed in our present setting. An example of application is given in constructing confidence intervals to estimate the criticality parameter in terms of log(\(\left( {{Z_{n + {n_0}}}/{Z_{{n_0}}}} \right)\)) and n.

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Acknowledgements

The authors are deeply indebted to the editor and the anonymous referee for their helpful comments. This work was supported by the National Natural Science Foundation of China (Grant Nos. 11601375, 11971063, 11731012), the Natural Science Foundation of Guangdong Province (Grant No. 2018A030313954), and the Centre Henri Lebesgue (CHL, ANR-11-LABX-0020-01).

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Authors and Affiliations

  1. Center for Applied Mathematics, Tian** University, Tian**, 300072, China

    **equan Fan

  2. School of Mathematics and Statistics, Northeastern University at Qinhuangdao, Qinhuangdao, 066004, China

    Haijuan Hu

  3. UMR CNRS 6205, Université de Bretagne-Sud, LMBA, Campus de Tohannic, 56017, Vannes, France

    Quansheng Liu

  4. School of Statistics and Mathematics, Zhongnan University of Economics and Law, Wuhan, 430073, China

    Quansheng Liu

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  1. **equan Fan
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  2. Haijuan Hu
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  3. Quansheng Liu
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Correspondence to **equan Fan.

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Fan, X., Hu, H. & Liu, Q. Uniform Cramér moderate deviations and Berry-Esseen bounds for a supercritical branching process in a random environment. Front. Math. China 15, 891–914 (2020). https://doi.org/10.1007/s11464-020-0868-3

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  • DOI: https://doi.org/10.1007/s11464-020-0868-3

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