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)β¦nβ1/2(Ξ²qr,sβ(K(n1/dSnβ))βE[Ξ²qr,sβ(K(n1/dSnβ))]). So far, pointwise limit theorems have been established in different
set-ups. 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 stabilizing 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),
0β€rβ€s<β, and that it is not affected by percolation effects in the
underlying random geometric graph