| 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
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| 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 Name | Affiliation | | 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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| Abstract: |
| 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. |
| Citation: |
| DOI:10.3770/j.issn:2095-2651.2023.06.009 |
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