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.

翻译：持久Betti数是拓扑数据分析子领域——持续同调理论中的核心工具。持续同调中的许多方法都依赖于将持久Betti数视为二维随机过程 $ (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))])$ 的性质。迄今为止，已在不同设定下建立了逐点极限定理。特别地，（持久）Betti数的逐点渐近正态性已在平稳泊松过程与强度函数恒定的二项过程中得到证明，该结果适用于所谓临界（或热力学）区域，参见Yogeshwaran等人[2017]与Hiraoka等人[2018]的研究。本文通过建立持久Betti数的强稳定性质（基于Penrose与Yukich[2001]的思想框架），将现有渐近正态性结果推广至多元情形及更广泛的泊松过程与二项过程类别。最重要的是，我们证明多元渐近正态性对所有参数对 $(r,s)$（其中 $0\le r\le s<\infty$）均成立，且该性质不受底层随机几何图中渗流效应的影响。