English

Limit theorems for random cubical homology

Probability 2018-03-20 v2 Algebraic Topology

Abstract

This paper studies random cubical sets in Rd\mathbb{R}^d. Given a cubical set XRdX\subset \mathbb{R}^d, a random variable ωQ[0,1]\omega_Q\in[0,1] is assigned for each elementary cube QQ in XX, and a random cubical set X(t)X(t) is defined by the sublevel set of XX consisting of elementary cubes with ωQt\omega_Q\leq t for each t[0,1]t\in[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.

Keywords

Cite

@article{arxiv.1612.08485,
  title  = {Limit theorems for random cubical homology},
  author = {Yasuaki Hiraoka and Kenkichi Tsunoda},
  journal= {arXiv preprint arXiv:1612.08485},
  year   = {2018}
}

Comments

19 pages, 14 figures

R2 v1 2026-06-22T17:34:47.550Z