On the asymptotic normality of persistent Betti numbers
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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) n^-1/2 (β^r,s_q ( K(n^1/d S_n))-E[β^r,s_q ( K( n^1/d S_n))]). So far, pointwise limit theorems have been established in different set-ups. 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 stabilizing 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< r< s<∞, and that it is not affected by percolation effects in the underlying random geometric graph.
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