Abstract
We study approximation theorems for the Euler characteristic of the Vietoris-Rips and Čech filtration. The filtration is obtained from a Poisson or binomial sampling scheme in the critical regime. We apply our results to the smooth bootstrap of the Euler characteristic and determine its rate of convergence in the Kantorovich-Wasserstein distance and in the Kolmogorov distance.
Highlights
In this manuscript we study approximation results and central limit theorems for the Euler characteristic (EC) χ of a simplicial complex K
Thomas and Owada [39] derive a functional strong law of large numbers and a functional central limit theorem (FCLT) for the EC obtained from the Vietoris-Rips complex of a Poisson process in the critical regime
Using the continuity properties of the Cech filtration, we extend the findings of [39] who provide a functional central limit theorem for the Vietoris-Rips complex and a Poisson sampling scheme
Summary
In this manuscript we study approximation results and central limit theorems for the Euler characteristic (EC) χ of a simplicial complex K. Thomas and Owada [39] derive a functional strong law of large numbers and a functional central limit theorem (FCLT) for the EC obtained from the Vietoris-Rips complex of a Poisson process in the critical regime. The underlying filtration (Kt,n : t ∈ [0, T ]) is the Cech or Vietoris-Rips complex of a Poisson or binomial point cloud in the critical regime We apply these results to a smooth bootstrap procedure proposed in Roycraft et al [36] and derive rates of convergence for the bootstrap procedure of the EC. Krebs and Polonik [25] established the strong stabilizing property of persistent Betti numbers and extended the validity of the central limit theorem to the binomial point process with a non-constant density.
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