  We study functional central limit theorems 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. The
results cover a directed sublevel-filtration for stabilizing networks and the
Cech and Vietoris-Rips complex on the random geometric graph.
  The presented functional central limit theorems open the door to a variety of
statistical applications in topological data analysis and we consider
goodness-of-fit tests in a simulation study.
