### 抄録

The persistent homology of a stationary point process on 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.

元の言語 | 英語 |
---|---|

ページ（範囲） | 2740-2780 |

ページ数 | 41 |

ジャーナル | Annals of Applied Probability |

巻 | 28 |

発行部数 | 5 |

DOI | |

出版物ステータス | 出版済み - 10 2018 |

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### All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Statistics, Probability and Uncertainty

### これを引用

*Annals of Applied Probability*,

*28*(5), 2740-2780. https://doi.org/10.1214/17-AAP1371

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

研究成果: ジャーナルへの寄稿 › 記事

*Annals of Applied Probability*, 巻. 28, 番号 5, pp. 2740-2780. https://doi.org/10.1214/17-AAP1371

}

TY - JOUR

T1 - Limit theorems for persistence diagrams

AU - Hiraoka, Yasuaki

AU - Shirai, Tomoyuki

AU - Trinh, Khanh Duy

PY - 2018/10

Y1 - 2018/10

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85052699592&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85052699592&partnerID=8YFLogxK

U2 - 10.1214/17-AAP1371

DO - 10.1214/17-AAP1371

M3 - Article

AN - SCOPUS:85052699592

VL - 28

SP - 2740

EP - 2780

JO - Annals of Applied Probability

JF - Annals of Applied Probability

SN - 1050-5164

IS - 5

ER -