Berry.Esseen theorems under weak dependence

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.