Sharp connections between Berry-Esseen characteristics and Edgeworth expansions for stationary processes
Sharp connections between Berry-Esseen characteristics and Edgeworth expansions for stationary processes
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DOI：
10.1090/tran/8328
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发表时间：
2020-07
影响因子：
1.3
通讯作者：
M. Jirak;W. Wu;Ou Zhao
中科院分区：
文献类型：
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作者：
M. Jirak;W. Wu;Ou Zhao
在给定一个弱相依平稳过程的情况下，我们用Berry-Essee特征描述了从Berry-Esseen界到二阶Edgeworth展开式之间的转变。这个特征是尖锐的：我们证明了Edgeworth展开式是有效的，当且仅当Berry-Essee特征具有一定的量级。如果不是这样，我们仍然得到一个最优的Berry-Esseen界，从而描述了确切的转变。我们还得到了给定$3<；p\leq 4$时刻的(分数)展开式，其中发生了类似的转变。相应的结果也适用于Wasserstein度量$W_1$，其中相关的综合特征被证明是最优的。作为应用，我们在$L^p$和$W_1$中建立了新的弱Edgeworth展开式和CLT。作为另一个应用，我们证明了一大类高维线性统计量在没有任何光滑性约束的情况下允许Edgeworth展开，也就是说，不需要非格点条件或相关条件。在所有结果中，必要的弱相依性假设是非常温和的。特别是，我们证明了时间序列分析中的许多重要的动力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.