LIMIT THEOREMS FOR PROCESS-LEVEL BETTI NUMBERS FOR SPARSE AND CRITICAL REGIMES

• Published in: Advances in applied probability Vol. 52; no. 1; pp. 1 - 31
• Main Authors: OWADA, TAKASHI, THOMAS, ANDREW M.
• Format: Journal Article
• Language: English
• Published: Sheffield Applied Probability Trust 01.03.2020; Cambridge University Press
• Subjects: Algebra; Asymptotic properties; Brownian motion; Central limit theorem; Connectivity; Constraining; Euclidean geometry; Gaussian process; Original Articles; Poisson density functions; Probability; Statistical analysis; Stochastic models; Stochastic processes; Theorems
• ISSN: 0001-8678; 1475-6064
• DOI: 10.1017/apr.2019.50

The objective of this study is to examine the asymptotic behavior of Betti numbers of Čech complexes treated as stochastic processes and formed from random points in the d-dimensional Euclidean space ℝ d . We consider the case where the points of the Čech complex are generated by a Poisson process with intensity nf for a probability density f. We look at the cases where the behavior of the connectivity radius of the Čech complex causes simplices of dimension greater than k + 1 to vanish in probability, the so-called sparse regime, as well when the connectivity radius is of the order of n -1/d , the critical regime. We establish limit theorems in the aforementioned regimes: central limit theorems for the sparse and critical regimes, and a Poisson limit theorem for the sparse regime. When the connectivity radius of the Čech complex is o(n -1/d ), i.e. the sparse regime, we can decompose the limiting processes into a time-changed Brownian motion or a time-changed homogeneous Poisson process respectively. In the critical regime, the limiting process is a centered Gaussian process but has a much more complicated representation, because the Čech complex becomes highly connected with many topological holes of any dimension.