关于相依数据下持久贝蒂数的极限定理
On limit theorems for persistent Betti numbers from dependent data
摘要：
研究了由时间序列和随机场得到的持久贝蒂数和持久图。众所周知，持久Betti函数是点云拓扑的有效描述符。到目前为止，$q$第th同调群的$(r,s)$ -持久贝蒂数$β^{r,s}_q$的收敛结果主要考虑从i.i.d.观测或平稳点过程，如泊松过程。在本文中，我们扩展了这些考虑。在相当一般的相依情形下，我们得到了持久贝蒂数$β^{r,s}_q$在临界区域的逐点收敛的极限定理.
英文摘要：
We study persistent Betti numbers and persistence diagrams obtained from time series and random fields. It is well known that the persistent Betti function is an efficient descriptor of the topology of a point cloud. So far, convergence results for the $(r,s)$-persistent Betti number of the $q$th homology group, $β^{r,s}_q$, were mainly considered for finite-dimensional point cloud data obtained from i.i.d. observations or stationary point processes such as a Poisson process. In this article, we extend these considerations. We derive limit theorems for the pointwise convergence of persistent Betti numbers $β^{r,s}_q$ in the critical regime under quite general dependence settings.
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