# Limit theorems for persistence diagrams

Yasuaki Hiraoka, Tomoyuki Shirai, Khanh Duy Trinh

Research output: Contribution to journalArticle

2 Citations (Scopus)

### Abstract

The persistent homology of a stationary point process on RN 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.

Original language English 2740-2780 41 Annals of Applied Probability 28 5 https://doi.org/10.1214/17-AAP1371 Published - 2018 Oct 1 Yes

### Fingerprint

Limit Theorems
Persistence
Diagram
Point Process
Homology
Continuum Percolation
Percolation Theory
Strong law of large numbers
Betti numbers
Stationary point
Stationary Process
Central limit theorem
Cavity
High-dimensional
Limiting
Infinity
Tend
Sufficient Conditions
Limit theorems
Diagrams

### Keywords

• Persistence diagram
• Persistent betti number
• Point process
• Random topology

### ASJC Scopus subject areas

• Statistics and Probability
• Statistics, Probability and Uncertainty

### Cite this

Limit theorems for persistence diagrams. / Hiraoka, Yasuaki; Shirai, Tomoyuki; Trinh, Khanh Duy.

In: Annals of Applied Probability, Vol. 28, No. 5, 01.10.2018, p. 2740-2780.

Research output: Contribution to journalArticle

Hiraoka, Yasuaki ; Shirai, Tomoyuki ; Trinh, Khanh Duy. / Limit theorems for persistence diagrams. In: Annals of Applied Probability. 2018 ; Vol. 28, No. 5. pp. 2740-2780.
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