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., 127(4), 1255-1281, 2017]. 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.

翻译：令 $(Z_{n})_{n\geq 0}$ 为独立同分布随机环境中的超临界分支过程。我们建立了该过程在Wasserstein-$1$距离下的最优收敛速率，完善了Grama等人[Stochastic Process. Appl., 127(4), 1255-1281, 2017]的结果。此外，还给出了指数型非一致Berry-Esseen界。最后，讨论了主要结果在临界参数和群体规模$Z_n$置信区间估计中的应用。