  Consider $(Z_n)_{n\geq0}$ 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 $(\log Z_n)_{n\geq0}$.
