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：持续Betti数是持续同调（拓扑数据分析的一个子领域）中的主要工具。持续同调中的许多方法依赖于将持续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，对于具有恒定强度函数的平稳泊松过程和二项式过程，在所谓的临界（或热力学）条件下，已建立了（持续）Betti数的逐点渐近正态性（参见Yogeshwaran等人 [2017] 和Hiraoka等人 [2018]）。在本研究中，我们推导了持续Betti数的强稳定性性质（遵循Penrose和Yukich [2001] 的思想），并将现有关于渐近正态性的结果推广到多元情形以及更广泛的底层泊松和二项式过程类别。最重要的是，我们证明了对所有满足$0\le r\le s<\infty$的配对$(r,s)$，多元渐近正态性均成立，且该性质不受底层随机几何图中渗流效应的影响。