LIMIT THEOREMS FOR PERSISTENCE DIAGRAMS

• 成果类型: Article
• 署名作者: Hiraoka, Yasuaki; Shirai, Tomoyuki; Khanh Duy Trinh
• 署名单位: Tohoku University; Kyushu University
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/17-AAP1371
• 发表日期: 2018
• 页码: 2740-2780
• 关键词: point-processes; TOPOLOGY

摘要

The persistent homology of a stationary point process on R-N is studied in this paper. As a generalization of continuum percolation theory, we study higher dimensional topological features of the point process such as loops, cavities, etc. in a multiscale way. The key ingredient is the persistence diagram, which is an expression of the persistent homology. We prove the strong law of large numbers for persistence diagrams as the window size tends to infinity and give a sufficient condition for the support of the limiting persistence diagram to coincide with the geometrically realizable region. We also discuss a central limit theorem for persistent Betti numbers.