关于持续 Betti 数的渐近正态性
On the asymptotic normality of persistent Betti numbers
AI总结：
本文证明持续 Betti 数具有强稳定化性质，并将其渐近正态性推广到多元情形及更广泛的 Poisson、二项过程，且结论不受渗流影响。
摘要：
持续 Betti 数是持续同调中的一种主要工具，而持续同调是拓扑数据分析的一个子领域。持续同调中的许多工具都依赖于将持续 Betti 数视为二维随机过程时所具有的性质，该过程为 $(r,s) \mapsto n^{-1/2}(β^{r,s}_q(\mathcal{K}(n^{1/d}\mathcal{X}_n))-\mathbb{E}[β^{r,s}_q(\mathcal{K}(n^{1/d}\mathcal{X}_n))])$。迄今为止，逐点极限定理已在不同设定下建立。特别是，对于具有常强度函数的平稳 Poisson 过程和二项过程，在所谓临界（或热力学）区域中，（持续）Betti 数的逐点渐近正态性已经得到建立，见 Yogeshwaran 等人 [2017] 以及 Hiraoka 等人 [2018]。在本文中，我们秉承 Penrose 和 Yukich [2001] 的精神，推导了持续 Betti 数的强稳定化性质，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一类底层 Poisson 过程和二项过程。最重要的是，我们证明多元渐近正态性对所有满足 $0\le r\le s<\infty$ 的参数对 $(r,s)$ 均成立，并且不受底层随机几何图中渗流效应的影响。
英文摘要：
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} (β^{r,s}_q ( \mathcal{K}(n^{1/d} \mathcal{X}_n))-\mathbb{E}[β^{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.
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