Limit theorems for unbounded cluster functionals of regularly varying time series
正则变动时间序列无界极端簇泛函的极限定理AI 解读 ↓
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正则变动时间序列无界极端簇泛函的极限定理
- 研究问题
- 平稳正则变动时间序列的极端簇，在使用无界或非平移不变泛函时具有怎样的极限行为，块长度又如何影响收敛速度与一致性？
- 方法与思路
- 作者通过分块定义极端簇及其点测度，研究不同泛函与块尺度下的极限，再分析经验簇测度的一致性和经验簇过程的弱收敛。
- 材料陈述的贡献
- 摘要称小块和中等块下经验簇测度具有一致性，大块下则失效；中等块和大块的经验簇过程须适当重新标准化才能弱收敛。
- 阅读关注点
- 应核对小、中、大块的精确定义、各类泛函的归一化速度，以及统一可积性和序列依赖条件如何决定一致性与弱收敛。
- 阅读边界
- 这是 arXiv 预印本，同行评审及期刊接收未核实；摘要未给出块尺度阈值、归一化公式或完整依赖条件，不能据此直接实施。
原文摘要
A blocks method is used to define clusters of extreme values in stationary time series. The cluster starts at the first large value in the block and ends at the last one. The block cluster measure (the point measure at clusters) encodes different aspects of extremal properties. Its limiting behaviour is handled by vague convergence, hence the set of test functions consists of bounded, shift-invariant functionals that vanish around zero. If unbounded or non shift-invariant functionals are considered, we may obtain convergence at a different rate, depending on the type of the functional and the block size (small vs. large blocks). There are two prominent examples of such functionals: the locations of large jumps and the cluster length. We obtain a comprehensive characterization of the limiting behaviour of the block cluster measure evaluated at such functionals for stationary, regularly varying time series. Once the convergence of the block cluster measure is established, we can proceed with consistency of the empirical cluster measure. Consistency holds in the small and moderate blocks scenario, while fails in the large blocks situation. Next, we continue with weak convergence of the empirical cluster processes. The starting point is the seminal paper by Drees and Rootzen (2010). Under the appropriate uniform integrability condition (related to small blocks) the results in the latter paper are still valid. In the moderate and large blocks scenario, the Drees and Rootzen empirical cluster process diverges, but converges weakly when re-normalized properly.