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 $3 < p \leq 4$ moments, where a similar transition occurs. Corresponding results also hold for the Wasserstein metric $W_1$, where a related, integrated characteristic turns out to be optimal. As an application, we establish novel weak Edgeworth expansion and CLTs in $L^p$ and $W_1$. As another application, we show that a large class of high dimensional linear statistics admit 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.

中文翻译：

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Berry-Esseen 特征与固定过程的 Edgeworth 展开之间的紧密联系

给定一个弱依赖的平稳过程，我们根据 Berry-Esseen 特性描述了 Berry-Esseen 界和二阶 Edgeworth 展开之间的过渡。这个特征很明显：我们证明了 Edgeworth 展开是有效的，当且仅当 Berry-Esseen 特征具有一定的量级。如果不是这种情况，我们仍然会得到一个最佳的 Berry-Esseen 界限，从而描述了确切的过渡。我们还获得了给定 $3 < p \leq 4$ 时刻的（分数）扩展，其中发生了类似的转换。相应的结果也适用于 Wasserstein 指标 $W_1$，其中相关的集成特性被证明是最佳的。作为一个应用程序，我们在 $L^p$ 和 $W_1$ 中建立了新颖的弱 Edgeworth 展开和 CLT。作为另一个应用程序，我们表明，一大类高维线性统计允许 Edgeworth 展开而没有任何平滑约束，也就是说，不需要非晶格条件或相关条件。在所有结果中，必要的弱依赖假设是非常温和的。特别是，我们展示了来自时间序列分析的许多突出的动力系统和模型都在我们的框架内，在这些领域产生了许多新的结果。