Berry–Esseen theorems under weak dependence

• Jirak, Moritz ^[1]
  1. [1] Humboldt University of Berlin

     Humboldt University of Berlin

     Berlin, Stadt, Alemania

     
• Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 44, Nº. 3, 2016, págs. 2024-2063
• Idioma: inglés
• DOI: 10.1214/15-AOP1017
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• Resumen

  • Let {Xk}k≥Z be a stationary sequence. Given p∈(2,3] moments and a mild weak dependence condition, we show a Berry–Esseen theorem with optimal rate np/2−1. For p≥4, we also show a convergence rate of n1/2 in Lq-norm, where q≥1. Up to logn 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.