Exact convergence rate of the central limit theorem and polynomial convergence rate for branching processes in a random environment

Abstract

Let (Zn) be a supercritical branching process in an independent and identically distributed (i.i.d.) random environment. The paper studies the properties of the estimator Mn=n−1∑k=0n−1(Zk+1/Zk) introduced by Dion and Esty in 1979. We introduce a related martingale and discuss its convergence and exponential convergence rate. On this basis the exact convergence rate of the central limit theorem for normalized Mn is given.