Functional central limit theorems for persistent Betti numbers on cylindrical networks

We study functional central limit theorems (FCLTs) for persistent Betti numbers obtained from networks defined on a Poisson point process. The limit is formed in large volumes of cylindrical shape stretching only in one dimension. Moreover, the limiting results cover two possible filtrations, namely a directed sublevel-filtration for stabilizing networks and the Vietoris-Rips complex on the random geometric graph. Finally, the presented FCLTs open the door to a variety of statistical applications in topological data analysis and we consider goodness-of-fit tests in a simulation study.