Berry-Esseen's bound and Cramér's large deviation expansion for a supercritical branching process in a random environment

Let (Zn) be a supercritical branching process in a random environment ξ=(ξn). We establish a Berry-Esseen bound and a Cramér's type large deviation expansion for logZn under the annealed law P. We also improve some earlier results about the harmonic moments of the limit variable W=limn→∞Wn, where Wn=Zn/EξZn is the normalized population size.