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On the asymptotic normality of persistent Betti numbers

Part of: Stochastic processes Limit theorems Geometric probability and stochastic geometry

Published online by Cambridge University Press:  03 December 2024

Johannes Krebs Open the ORCID record for Johannes Krebs [Opens in a new window]  and
Wolfgang Polonik
Johannes Krebs*
Affiliation:
KU Eichstätt-Ingolstadt
Wolfgang Polonik*
Affiliation:
University of California at Davis
*
*Postal address: KU Eichstätt-Ingolstadt, Ostenstraße 28, 85072 Eichstätt, Germany. Email address: [email protected]
**Postal address: Department of Statistics, University of California, Davis, CA 95616, USA. Email address: [email protected]
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Abstract

Persistent Betti numbers are a major tool in persistent homology, a subfield of topological data analysis. Many tools in persistent homology rely on the properties of persistent Betti numbers considered as a two-dimensional stochastic process $ (r,s) \mapsto n^{-1/2} (\beta^{r,s}_q ( \mathcal{K}(n^{1/d} \mathcal{X}_n))-\mathbb{E}[\beta^{r,s}_q ( \mathcal{K}( n^{1/d} \mathcal{X}_n))])$. So far, pointwise limit theorems have been established in various settings. In particular, the pointwise asymptotic normality of (persistent) Betti numbers has been established for stationary Poisson processes and binomial processes with constant intensity function in the so-called critical (or thermodynamic) regime; see Yogeshwaran et al. (Prob. Theory Relat. Fields 167, 2017) and Hiraoka et al. (Ann. Appl. Prob. 28, 2018).

In this contribution, we derive a strong stabilization property (in the spirit of Penrose and Yukich, Ann. Appl. Prob. 11, 2001) of persistent Betti numbers, and we generalize the existing results on their asymptotic normality to the multivariate case and to a broader class of underlying Poisson and binomial processes. Most importantly, we show that multivariate asymptotic normality holds for all pairs (r, s), $0\le r\le s<\infty$, and that it is not affected by percolation effects in the underlying random geometric graph.

Keywords

Critical regimestrong stabilizationtopological data analysisrandom geometric complexesweak convergence

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry 60F05: Central limit and other weak theorems
Secondary: 60G55: Point processes
Type
Original Article
Information
Advances in Applied Probability , Volume 57 , Issue 2 , June 2025 , pp. 492 - 523
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Applied Probability Trust

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