On the asymptotic normality of persistent Betti numbers

TL;DR

This paper proves the multivariate asymptotic normality of persistent Betti numbers in topological data analysis, extending previous pointwise results to broader settings and showing robustness against percolation effects.

Contribution

It introduces a strong stabilization property for persistent Betti numbers and generalizes asymptotic normality results to multivariate cases and more general Poisson and binomial processes.

Findings

Multivariate asymptotic normality holds for all pairs (r,s) in the specified range.

Results are robust to percolation effects in the underlying graph.

Extends previous pointwise theorems to broader classes of processes.

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 different 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. [2017] and Hiraoka et al. [2018]. In this contribution, we derive a strong stabilization property (in the spirit of Penrose and Yukich [2001] of persistent Betti numbers and generalize the existing results on the asymptotic normality to the multivariate case and to a broader class of underlying Poisson and binomial processes. Most importantly, we show that the 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.