论文标题
浆果 - Esseen特征与Edgeworth扩展之间的急剧连接
Sharp connections between Berry-Esseen characteristics and Edgeworth expansions for stationary processes
论文作者
论文摘要
给定一个弱依赖的固定过程，我们描述了浆果 - 埃斯尼的结合与二阶阶程Edgeworth扩展之间的过渡。这种特征是敏锐的：我们表明，只有当浆果 - 埃森特征具有一定程度的情况下，Edgeworth的扩展是有效的。如果不是这种情况，我们仍然会得到一个最佳的浆果 - 埃斯尼结合，从而描述了确切的过渡。我们还获得了（分数）扩展，给出了$ 3 <p \ leq 4 $矩，其中发生了类似的过渡。相应的结果也适用于Wasserstein Metric $ W_1 $，其中相关的集成特征证明是最佳的。作为一个应用程序，我们在$ l^p $和$ w_1 $中建立了新颖的弱Edgeworth扩展和CLT。作为另一个应用程序，我们表明，一大批高维线性统计量会允许Edgeworth扩展而没有任何平滑度约束，即无需非务实条件或相关。在所有结果中，必要的弱依赖性假设非常温和。特别是，我们表明，时间序列分析的许多突出的动态系统和模型都在我们的框架内，从而在这些领域产生了许多新的结果。
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.
