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

doi:10.1007/s11464-020-0868-3

Front. Math. China

2020, Vol. 15

Issue (5) : 891-914 https://doi.org/10.1007/s11464-020-0868-3

RESEARCH ARTICLE

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

Xiequan FAN¹(

), Haijuan HU², Quansheng LIU^3,⁴

¹. Center for Applied Mathematics, Tianjin University, Tianjin 300072, China
². School of Mathematics and Statistics, Northeastern University at Qinhuangdao, Qinhuangdao 066004, China
³. Université de Bretagne-Sud, LMBA, UMR CNRS 6205, Campus de Tohannic, 56017 Vannes, France
⁴. School of Statistics and Mathematics, Zhongnan University of Economics and Law, Wuhan 430073, China

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Abstract

Let ${Z n, n ≥ 0}$ be a supercritical branching process in an independent and identically distributed random environment. We prove Cramér moderate deviations and Berry-Esseen bounds for log( $Z n + n 0 / Z n 0$ ) uniformly in $n 0 ∈ ℤ$ ,which extend the corresponding results by I. Grama, Q. Liu, and M. Miqueu [Stochastic Process. Appl., 2017, 127: 1255–1281] established for $n 0 = 0$ . 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( $Z n + n 0 / Z n 0$ ) and n.

Keywords Branching processes random environment Cramér moderate deviations Berry-Esseen bounds

Corresponding Author(s): Xiequan FAN

Issue Date: 19 November 2020

Cite this article:

Xiequan FAN,Haijuan HU,Quansheng LIU. Uniform Cramér moderate deviations and Berry-Esseen bounds for a supercritical branching process in a random environment[J]. Front. Math. China, 2020, 15(5): 891-914.

URL:

https://academic.hep.com.cn/fmc/EN/10.1007/s11464-020-0868-3
https://academic.hep.com.cn/fmc/EN/Y2020/V15/I5/891

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