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

Abstract

For a supercritical branching process ( Z n ) in a stationary and ergodic environment ξ , we study the rate of convergence of the normalized population W n = Z n / E [ Z n | ξ ] to its limit W ∞ : we show a central limit theorem for W ∞ − W n 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 W n + k − W n for each fixed k ∈ N ∗ .