From law of the iterated logarithm to Zolotarev distance for supercritical branching processes in random environment

Abstract

Consider ${\left(Z_n\right)}_{n⩾0}$ a supercritical branching process in an independent and identically distributed environment. Based on some recent development in martingale limit theory, we established law of the iterated logarithm, strong law of large numbers, invariance principle and optimal convergence rate in the central limit theorem under Zolotarev and Wasserstein distances of order $p\in (0,2]$ for the process ${(log{Z}_n)}_{n⩾0}$.