Limit theorems for persistence diagrams

Yasuaki Hiraoka, Tomoyuki Shirai, Khanh Duy Trinh

Research output: Contribution to journalArticlepeer-review

33 Citations (Scopus)


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 languageEnglish
Pages (from-to)2740-2780
Number of pages41
JournalAnnals of Applied Probability
Issue number5
Publication statusPublished - 2018 Oct
Externally publishedYes


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

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


Dive into the research topics of 'Limit theorems for persistence diagrams'. Together they form a unique fingerprint.

Cite this