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On limit theorems for persistent Betti numbers from dependent data

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  • Krebs, Johannes

Abstract

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 qth homology group, βqr,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 βqr,s in the critical regime under quite general dependence settings.

Suggested Citation

  • Krebs, Johannes, 2021. "On limit theorems for persistent Betti numbers from dependent data," Stochastic Processes and their Applications, Elsevier, vol. 139(C), pages 139-174.
    • Handle: RePEc:eee:spapps:v:139:y:2021:i:c:p:139-174
    DOI: 10.1016/j.spa.2021.04.013
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    References listed on IDEAS

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    1. Marian Gidea, 2017. "Topology data analysis of critical transitions in financial networks," Papers 1701.06081, arXiv.org.
    2. Umar Islambekov & Monisha Yuvaraj & Yulia R. Gel, 2020. "Harnessing the power of topological data analysis to detect change points," Environmetrics, John Wiley & Sons, Ltd., vol. 31(1), February.
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