We study functional central limit theorems for persistent Betti numbers obtained from networks defined on a Poisson point process. The limit is formed in large volumes of cylindrical shape stretching only in one dimension. The results cover a directed sublevel-filtration for stabilizing networks and the Čech and Vietoris–Rips complex on the random geometric graph. The presented functional central limit theorems open the door to a variety of statistical applications in topological data analysis and we consider goodness-of-fit tests in a simulation study.

中文翻译：

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圆柱网络上持久 Betti 数的函数中心极限定理

我们研究了从泊松点过程定义的网络中获得的持久 Betti 数的功能中心极限定理。极限是在仅在一维上拉伸的大量圆柱形状中形成的。结果涵盖了用于稳定网络的定向子级过滤以及随机几何图上的 Čech 和 Vietoris-Rips 复数。所提出的功能中心极限定理为拓扑数据分析中的各种统计应用打开了大门，我们在模拟研究中考虑了拟合优度检验。