Berry–Esseen theorems under weak dependence

Abstract

Let {Xk}k≥Z be a stationary sequence. Given p∈(2,3] moments and a mild weak dependence condition, we show a Berry–Esseen theorem with optimal rate np/2−1. For p≥4, we also show a convergence rate of n1/2 in Lq-norm, where q≥1. Up to logn factors, we also obtain nonuniform rates for any p>2. This leads to new optimal results for many linear and nonlinear processes from the time series literature, but also includes examples from dynamical system theory. The proofs are based on a hybrid method of characteristic functions, coupling and conditioning arguments and ideal metrics.