Berry-Esseen theorems under weak dependence

TL;DR

This paper establishes optimal Berry-Esseen theorems for stationary sequences with weak dependence, providing convergence rates in distribution and in $ ext{L}^q$-norm for various moment conditions, applicable to time series and dynamical systems.

Contribution

It introduces new Berry-Esseen bounds under weak dependence with optimal rates, extending classical results to broader classes of processes.

Findings

Optimal rate $n^{p/2-1}$ for Berry-Esseen theorem under weak dependence.

Convergence rate of $n^{1/2}$ in $ ext{L}^q$-norm for $p extgreater 4$.

Nonuniform rates up to $ ext{log} n$ factors for $p extgreater 2$.

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.