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October 2018 Limit theorems for persistence diagrams
Yasuaki Hiraoka, Tomoyuki Shirai, Khanh Duy Trinh
Ann. Appl. Probab. 28(5): 2740-2780 (October 2018). DOI: 10.1214/17-AAP1371


The persistent homology of a stationary point process on $\mathbf{R}^{N}$ is studied in this paper. As a generalization of continuum percolation theory, we study higher dimensional topological features of the point process such as loops, cavities, etc. in a multiscale way. The key ingredient is the persistence diagram, which is an expression of the persistent homology. We prove the strong law of large numbers for persistence diagrams as the window size tends to infinity and give a sufficient condition for the support of the limiting persistence diagram to coincide with the geometrically realizable region. We also discuss a central limit theorem for persistent Betti numbers.


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Yasuaki Hiraoka. Tomoyuki Shirai. Khanh Duy Trinh. "Limit theorems for persistence diagrams." Ann. Appl. Probab. 28 (5) 2740 - 2780, October 2018.


Received: 1 December 2016; Revised: 1 August 2017; Published: October 2018
First available in Project Euclid: 28 August 2018

zbMATH: 06974764
MathSciNet: MR3847972
Digital Object Identifier: 10.1214/17-AAP1371

Primary: 60B10 , 60K35
Secondary: 55N20

Keywords: persistence diagram , persistent Betti number , point process , random topology

Rights: Copyright © 2018 Institute of Mathematical Statistics


Vol.28 • No. 5 • October 2018
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