  We study persistent Betti numbers and persistence diagrams obtained from time
series and random fields. It is well known that the persistent Betti function
is an efficient descriptor of the topology of a point cloud. So far,
convergence results for the $(r,s)$-persistent Betti number of the $q$th
homology group, $\beta^{r,s}_q$, were mainly considered for finite-dimensional
point cloud data obtained from i.i.d. observations or stationary point
processes such as a Poisson process. In this article, we extend these
considerations. We derive limit theorems for the pointwise convergence of
persistent Betti numbers $\beta^{r,s}_q$ in the critical regime under quite
general dependence settings.
