The Annals of Applied Probability

Limit theorems for persistence diagrams

Yasuaki Hiraoka, Tomoyuki Shirai, and Khanh Duy Trinh

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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.

Article information

Ann. Appl. Probab., Volume 28, Number 5 (2018), 2740-2780.

Received: December 2016
Revised: August 2017
First available in Project Euclid: 28 August 2018

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Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60B10: Convergence of probability measures
Secondary: 55N20: Generalized (extraordinary) homology and cohomology theories

Point process persistence diagram persistent Betti number random topology


Hiraoka, Yasuaki; Shirai, Tomoyuki; Trinh, Khanh Duy. Limit theorems for persistence diagrams. Ann. Appl. Probab. 28 (2018), no. 5, 2740--2780. doi:10.1214/17-AAP1371.

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