Wasserstein-$1$ Distance and Nonuniform Berry-Esseen Bound for a Supercritical Branching Process in a Random Environment
Received:September 29, 2022  Revised:June 01, 2023
Key Words: Branching processes   Random environment   Wasserstein-$1$ distance   Nonuniform Berry-Esseen bounds  
Fund Project:Supported by the National Natural Science Foundation of China (Grant No.11971063) and CY Initiative of Excellence (Grant No.``Investissements d'Avenir" ANR-16-IDEX-0008), Project ``EcoDep" (Grant No.PSI-AAP2020-0000000013).
Author NameAffiliation
Hao WU Center for Applied Mathematics, Tianjin University, Tianjin 300072, P. R. China 
Xiequan FAN School of Mathematics and Statistics, Northeastern University at Qinhuangdao, Hebei 066004, P. R. China 
Zhiqiang GAO Laboratory of Mathematics and Complex Systems, School of Mathematical Sciences, Beijing Normal University, Beijing 100875, P. R. China 
Yinna YE Department of Applied Mathematics, School of Mathematics and Physics, Xi'an Jiaotong-Liverpool University, Jiangsu 215123, P. R. China 
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      Let $(Z_{n})_{n\geq 0}$ be a supercritical branching process in an independent and identically distributed random environment. We establish an optimal convergence rate in the Wasserstein-$1$ distance for the process $(Z_{n})_{n\geq 0}$, which completes a result of Grama et al. [Stochastic Process. Appl., 2017, \textbf{127}(4): 1255--1281]. Moreover, an exponential nonuniform Berry-Esseen bound is also given. At last, some applications of the main results to the confidence interval estimation for the criticality parameter and the population size $Z_n$ are discussed.
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