Limit theorems for process-level Betti numbers for sparse and critical regimes
| Autor: | Andrew M. Thomas, Takashi Owada |
|---|---|
| Rok vydání: | 2020 |
| Předmět: |
Statistics and Probability
Pure mathematics Stochastic process Betti number Euclidean space Applied Mathematics 010102 general mathematics 01 natural sciences 010104 statistics & probability symbols.namesake symbols Limit (mathematics) 0101 mathematics Gaussian process Brownian motion Central limit theorem Mathematics Poisson limit theorem |
| Zdroj: | Advances in Applied Probability. 52:1-31 |
| ISSN: | 1475-6064 0001-8678 |
| Popis: | 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 ${\mathbb{R}}^d$ . 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. |
| Databáze: | OpenAIRE |
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