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

翻译：我们研究Vietoris-Rips和Cech过滤的Euler特征近似理论,过滤来自关键政体的Poisson或二进制取样方法,我们把结果应用于Euler特征的光滑靴子,并确定其在Kantorovich-Wasserstein距离和Kolmogorov距离上的趋同率。