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, ${\beta }_q^{r,s}$, 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 ${\beta }_q^{r,s}$ in the critical regime under quite general dependence settings.

我们研究从时间序列和随机字段获得的持久Betti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，$（\mathrm{[R}，s）$的持久性Betti数 $q$同源组 ${\beta }_q^{\mathrm{[R}，s}$，主要考虑从iid观测或平稳点过程（例如泊松过程）获得的有限维点云数据。在本文中，我们扩展了这些注意事项。我们导出了持久Betti数的逐点收敛的极限定理${\beta }_q^{\mathrm{[R}，s}$ 在相当普遍的依赖条件下的关键政权中。