Berry-Esseen theorems under weak dependence

Abstract

Let $\{X_k{\}}_{k\ge \mathbb{Z}}$ 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 $p\ge 4$, we also show a convergence rate of $n^{1/2}$ in ${\mathcal{L}}^q$-norm, where $q\ge 1$. 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.

Published at http://dx.doi.org/10.1214/15-AOP1017 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org). Minor corrections, results remain unchanged. Special thanks to Florence Merlevede, for pointing out some errors

References in corpus (3)

• Normal approximation under local dependence
• Komlós-Major-Tusnády approximation under dependence
• On mean central limit theorems for stationary sequences

Cited by in corpus (4)

• Central limit theorems for high dimensional dependent data
• Bagged filters for partially observed interacting systems
• Convergence rates in the central limit theorem for weighted sums of Bernoulli random fields
• U-statistics of local sample moments under weak dependence