Central limit theorems for a supercritical branching process in a random environment

For a supercritical branching process (Zn) in a stationary and ergodic environment ξ, we study the rate of convergence of the normalized population Wn=Zn/E[Zn|ξ] to its limit W∞: we show a central limit theorem for W∞−Wn with suitable normalization and derive a Berry-Esseen bound for the rate of convergence in the central limit theorem when the environment is independent and identically distributed. Similar results are also shown for Wn+k−Wn for each fixed k∈N∗.