LIMIT THEOREMS FOR POINT PROCESSES UNDER GEOMETRIC CONSTRAINTS (AND TOPOLOGICAL CRACKLE)

• 成果类型: Article
• 署名作者: Owada, Takashi; Adler, Robert J.
• 署名单位: Technion Israel Institute of Technology; Technion Israel Institute of Technology
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/16-AOP1106
• 发表日期: 2017
• 页码: 2004-2055
• 关键词: random-variables; HOMOLOGY

摘要

We study the asymptotic nature of geometric structures formed from a point cloud of observations of (generally heavy tailed) distributions in a Euclidean space of dimension greater than one. A typical example is given by the Betti numbers of Cech complexes built over the cloud. The structure of dependence and sparcity (away from the origin) generated by these distributions leads to limit laws expressible via nonhomogeneous, random, Poisson measures. The parametrisation of the limits depends on both the tail decay rate of the observations and the particular geometric constraint being considered. The main theorems of the paper generate a new class of results in the well established theory of extreme values, while their applications are of significance for the fledgling area of rigorous results in topological data analysis. In particular, they provide a broad theory for the empirically well-known phenomenon of homological crackle; the continued presence of spurious homology in samples of topological structures, despite increased sample size.