Journal article
LIMIT THEOREMS FOR PROCESS-LEVEL BETTI NUMBERS FOR SPARSE AND CRITICAL REGIMES
Advances in applied probability, Vol.52(1), pp.1-31
03/01/2020
DOI: 10.1017/apr.2019.50
Abstract
The objective of this study is to examine the asymptotic behavior of Betti numbers of Cech 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 Cech 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 Cech 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 Cech 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 Cech complex becomes highly connected with many topological holes of any dimension.
Details
- Title: Subtitle
- LIMIT THEOREMS FOR PROCESS-LEVEL BETTI NUMBERS FOR SPARSE AND CRITICAL REGIMES
- Creators
- Takashi Owada - Purdue University SystemAndrew M. Thomas - Purdue University System
- Resource Type
- Journal article
- Publication Details
- Advances in applied probability, Vol.52(1), pp.1-31
- Publisher
- Applied Probability Trust
- DOI
- 10.1017/apr.2019.50
- ISSN
- 0001-8678
- eISSN
- 1475-6064
- Number of pages
- 31
- Grant note
- 1811428 / National Science Foundation (NSF) grant, Division of Mathematical Science (DMS); National Science Foundation (NSF)
- Language
- English
- Date published
- 03/01/2020
- Academic Unit
- Statistics and Actuarial Science
- Record Identifier
- 9984446559202771
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