On approximation theorems for the Euler characteristic with applications to the bootstrap

We study approximation theorems for the Euler characteristic of the Vietoris-Rips and \v Cech 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.