Wasserstein-$1$ distance and nonuniform Berry-Esseen bound for a supercritical branching process in a random environment

Let $(Z_n{)}_{n\ge 0}$ 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 $(Z_n{)}_{n\ge 0}$, 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.