## The Annals of Applied Probability

### Limit theorems for persistence diagrams

#### Abstract

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

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

Dates
Revised: August 2017
First available in Project Euclid: 28 August 2018

https://projecteuclid.org/euclid.aoap/1535443233

Digital Object Identifier
doi:10.1214/17-AAP1371

Mathematical Reviews number (MathSciNet)
MR3847972

Zentralblatt MATH identifier
06974764

#### Citation

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. https://projecteuclid.org/euclid.aoap/1535443233

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