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May 2016 Berry–Esseen theorems under weak dependence
Moritz Jirak
Ann. Probab. 44(3): 2024-2063 (May 2016). DOI: 10.1214/15-AOP1017

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.

Citation

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Moritz Jirak. "Berry–Esseen theorems under weak dependence." Ann. Probab. 44 (3) 2024 - 2063, May 2016. https://doi.org/10.1214/15-AOP1017

Information

Received: 1 July 2014; Revised: 1 March 2015; Published: May 2016
First available in Project Euclid: 16 May 2016

zbMATH: 1347.60011
MathSciNet: MR3502600
Digital Object Identifier: 10.1214/15-AOP1017

Subjects:
Primary: 60F05

Keywords: Berry–Esseen , stationary process , Weak dependence

Rights: Copyright © 2016 Institute of Mathematical Statistics

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Vol.44 • No. 3 • May 2016
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