{
  "id": "1910.00751",
  "version": "v1",
  "published": "2019-10-02T02:37:37.000Z",
  "updated": "2019-10-02T02:37:37.000Z",
  "title": "Functional limit theorems for the Euler characteristic process in the critical regime",
  "authors": [
    "Andrew M. Thomas",
    "Takashi Owada"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "This study presents functional limit theorems for the Euler characteristic of Vietoris-Rips complexes. The points are drawn from a non-homogeneous Poisson process on $\\mathbb{R}^d$, and the connectivity radius governing the formation of simplices is taken as a function of time parameter $t$, which allows us to treat the Euler characteristic as a stochastic process. The setting in which this takes place is that of the critical regime, in which the simplicial complexes are highly connected and have non-trivial topology. We establish two \"functional-level\" limit theorems, a strong law of large numbers and a central limit theorem for the appropriately normalized Euler characteristic process.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-10-02T02:37:37.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60F17",
      "55U10",
      "60C05",
      "60D05"
    ],
    "keywords": [
      "functional limit theorems",
      "critical regime",
      "central limit theorem",
      "appropriately normalized euler characteristic process",
      "vietoris-rips complexes"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}