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

Given a weakly dependent stationary process, we describe the transition between Berry-Esseen bound and second order Edgeworth expansion in terms of characteristic. This characteristic is sharp: We show that expansions are valid if only certain magnitude. If this not case, still get an optimal bound, thus describing exact transition. also obtain (fractional) given 3 > p ? 4 3 > \leq 4 moments, where similar occurs. Corresponding results hold for Wasserstein metric alttext="upper W 1"> W 1 encoding="application/x-tex">W_1 , related, integrated turns out to be optimal. As application, establish novel weak CLTs L Superscript p"> L encoding="application/x-tex">L^p . another large class high dimensional linear statistics admits without any smoothness constraints, is, no non-lattice condition or related necessary. In all results, necessary weak-dependence assumptions very mild. particular, many prominent dynamical systems models from time series analysis within our framework, giving rise new these areas.