几何约束（和拓扑裂纹）下点过程的极限定理
Limit Theorems for Point Processes under Geometric Constraints (and Topological Crackle)
摘要：
我们研究了几何结构的渐近性质形成的观察点云（一般重尾）分布在欧几里得空间的尺寸大于1。一个典型的例子是建立在云上的Čech复形的贝蒂数。由这些分布产生的依赖性和稀疏性（远离原点）的结构导致了通过非齐次、随机、泊松测度可表达的极限定律。极限的参数化取决于观测的尾部衰减率和所考虑的特定几何约束。本文的主要定理在已建立的极值理论中产生了一类新的结果，而它们的应用对于拓扑数据分析中的严格结果的新兴领域具有重要意义。特别是，他们提供了一个广泛的理论，经验上众所周知的现象同源“裂纹”，继续存在虚假的同源性的拓扑结构的样本，尽管增加了样本量。
英文摘要：
We study the asymptotic nature of geometric structures formed from a point cloud of observations of (generally heavy tailed) distributions in a Euclidean space of dimension greater than one. A typical example is given by the Betti numbers of Čech complexes built over the cloud. The structure of dependence and sparcity (away from the origin) generated by these distributions leads to limit laws expressible via non-homogeneous, random, Poisson measures. The parametrisation of the limits depends on both the tail decay rate of the observations and the particular geometric constraint being considered. The main theorems of the paper generate a new class of results in the well established theory of extreme values, while their applications are of significance for the fledgling area of rigorous results in topological data analysis. In particular, they provide a broad theory for the empirically well-known phenomenon of homological `crackle'; the continued presence of spurious homology in samples of topological structures, despite increased sample size.
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