AI 中文总结
研究Germ-Grain随机集模型中持久图泛函的中心极限定理，基于标记点过程稳定化方法，证明特定条件下其渐近正态性，用于拟合优度检验区分模型，还应用于乳腺组织学图像。
AI 中文摘要
本文建立了源于Germ-Grain随机集模型的M-有界持久图泛函的中心极限定理。基于标记点过程的稳定化方法，表明在特定条件下，随着观测窗口增加，这些拓扑摘要呈现渐近正态性，尤其对于具有指数相关衰减的模型。这些结果应用于检测聚类或排斥等空间相互作用的拟合优度检验。利用从持久图的矩形划分和泛函摘要导出的检验统计量区分不同模型。最后将该方法应用于乳腺组织的组织学图像。
英文摘要
This paper establishes a central limit theorem (CLT) for functionals of $M$-bounded persistence diagrams arising from germ-grain random set models. Building on stabilisation methods for marked point processes, we show that, under certain conditions, these topological summaries exhibit asymptotic normality as the observation window increases, particularly for models with exponential decay of correlations. These results are applied in goodness-of-fit tests designed to detect spatial interactions such as clustering or repulsion. Using test statistics derived from rectangular partitions of persistence diagrams and functional summaries (e.g., the APF or the support function of the lift zonoid), the study distinguishes between different models. Finally, the methodology is applied to histological images of breast tissue.
