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

Abstract

Let $\left(Z_n\right)$ be a supercritical branching process in an independent and identically distributed (i.i.d.) random environment. The paper studies the properties of the estimator $M_n=n^{-1}{\sum }_{k=0}^{n-1}(Z_{k+1}/Z_k)$ 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 $M_n$ is given.