  Persistent Betti numbers are a major tool in persistent homology, a subfield
of topological data analysis. Many tools in persistent homology rely on the
properties of persistent Betti numbers considered as a two-dimensional
stochastic process $ (r,s) \mapsto n^{-1/2} (\beta^{r,s}_q (
\mathcal{K}(n^{1/d} \mathcal{X}_n))-\mathbb{E}[\beta^{r,s}_q ( \mathcal{K}(
n^{1/d} \mathcal{X}_n))])$. So far, pointwise limit theorems have been
established in different settings. In particular, the pointwise asymptotic
normality of (persistent) Betti numbers has been established for stationary
Poisson processes and binomial processes with constant intensity function in
the so-called critical (or thermodynamic) regime, see Yogeshwaran et al. [2017]
and Hiraoka et al. [2018].
  In this contribution, we derive a strong stabilization property (in the
spirit of Penrose and Yukich [2001] of persistent Betti numbers and generalize
the existing results on the asymptotic normality to the multivariate case and
to a broader class of underlying Poisson and binomial processes. Most
importantly, we show that the multivariate asymptotic normality holds for all
pairs $(r,s)$, $0\le r\le s<\infty$, and that it is not affected by percolation
effects in the underlying random geometric graph.
