Limit Theorems for Random Cubical Homology

Yasuaki Hiraoka, Kenkichi Tsunoda

Research output: Contribution to journalArticlepeer-review

3 Citations (Scopus)

Abstract

This paper studies random cubical sets in Rd. Given a cubical set X⊂ Rd, a random variable ωQ∈ [ 0 , 1 ] is assigned for each elementary cube Q in X, and a random cubical set X(t) is defined by the sublevel set of X consisting of elementary cubes with ωQ≤ t for each t∈ [ 0 , 1 ]. Under this setting, the main results of this paper show the limit theorems (law of large numbers and central limit theorem) for Betti numbers and lifetime sums of random cubical sets and filtrations. In addition to the limit theorems, the positivity of the limiting Betti numbers is also shown.

Original languageEnglish
Pages (from-to)665-687
Number of pages23
JournalDiscrete and Computational Geometry
Volume60
Issue number3
DOIs
Publication statusPublished - 2018 Oct 1

Keywords

  • Betti number
  • Cubical complex
  • Cubical homology
  • Random topology

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Geometry and Topology
  • Discrete Mathematics and Combinatorics
  • Computational Theory and Mathematics

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