Sharp connections between Berry-Esseen characteristics and Edgeworth expansions for stationary processes

Johannes Moritz Jirak, Wei Biao Wu, Ou Zhao

Publications: Contribution to journalArticlePeer Reviewed

Abstract

Given a weakly dependent stationary process, we describe the transition between a Berry-Esseen bound and a second order Edgeworth expansion in terms of the Berry-Esseen characteristic. This characteristic is sharp: We show that Edgeworth expansions are valid if and only if the Berry-Esseen characteristic is of a certain magnitude. If this is not the case, we still get an optimal Berry-Esseen bound, thus describing the exact transition. We also obtain (fractional) expansions given moments, where a similar transition occurs. Corresponding results also hold for the Wasserstein metric , where a related, integrated characteristic turns out to be optimal. As an application, we establish novel weak Edgeworth expansion and CLTs in and . As another application, we show that a large class of high dimensional linear statistics admits Edgeworth expansions without any smoothness constraints, that is, no non-lattice condition or related is necessary. In all results, the necessary weak-dependence assumptions are very mild. In particular, we show that many prominent dynamical systems and models from time series analysis are within our framework, giving rise to many new results in these areas.
Original languageEnglish
Pages (from-to)4129-4183
Number of pages55
JournalTransactions of the American Mathematical Society
Volume374
Issue number6
Early online date19 Mar 2021
DOIs
Publication statusPublished - 2021

Austrian Fields of Science 2012

  • 101019 Stochastics

Keywords

  • AGGREGATION
  • ASYMPTOTIC EXPANSIONS
  • BOUNDS
  • Berry-Esseen
  • Berry-Esseen characteristic
  • CENTRAL-LIMIT-THEOREM
  • CONVERGENCE
  • DISTRIBUTIONS
  • Edgeworth expansions
  • FIELDS
  • PRODUCTS
  • SUMS
  • exact transition
  • weak dependence
  • Weak dependence
  • Exact transition

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