Berry–Esseen theorems under weak dependence

Abstract

Let $\{{X}_{k}\}_{k\geq\mathbb{Z}}$ be a stationary sequence. Given $p\in(2,3]$ moments and a mild weak dependence condition, we show a Berry–Esseen theorem with optimal rate $n^{p/2-1}$. For $p\geq4$, we also show a convergence rate of $n^{1/2}$ in $\mathcal{L}^{q}$-norm, where $q\geq1$. Up to $\log n$ 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.