BERRY-ESSEEN THEOREMS UNDER WEAK DEPENDENCE

• 成果类型: Article
• 署名作者: Jirak, Moritz
• 署名单位: Humboldt University of Berlin
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/15-AOP1017
• 发表日期: 2016
• 页码: 2024-2063
• 关键词: CENTRAL-LIMIT-THEOREM; asymptotic expansions; normal approximation; steins method; CONVERGENCE; bounds; statistics; FIELDS; rates; sums

摘要

Let {X-k}(k >= Z) be a stationary sequence. Given p is an element of (2, 3] moments and a mild weak dependence condition, we show a Berry-Esseen theorem with optimal rate n(p/2-1). For p >= 4, we also show a convergence rate of n(1/2) in L-q-norm, where q >= 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.