{
  "id": "2103.05799",
  "version": "v1",
  "published": "2021-03-10T00:54:48.000Z",
  "updated": "2021-03-10T00:54:48.000Z",
  "title": "Functional strong law of large numbers for Betti numbers in the tail",
  "authors": [
    "Takashi Owada",
    "Zifu Wei"
  ],
  "comment": "43 pages, 2 figures",
  "categories": [
    "math.PR"
  ],
  "abstract": "The objective of this paper is to investigate the layered structure of topological complexity in the tail of a probability distribution. We establish the functional strong law of large numbers for Betti numbers, a basic quantifier of algebraic topology, of a geometric complex outside an open ball of radius $R_n$, such that $R_n\\to\\infty$ as the sample size $n$ increases. The nature of the obtained law of large numbers is determined by the decay rate of a probability density. It especially depends on whether the tail of a density decays at a regularly varying rate or an exponentially decaying rate. The nature of the limit theorem depends also on how rapidly $R_n$ diverges. In particular, if $R_n$ diverges sufficiently slowly, the limiting function in the law of large numbers is crucially affected by the emergence of arbitrarily large connected components supporting topological cycles in the limit.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-03-10T00:54:48.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G70",
      "60F15",
      "55U10",
      "60D05",
      "60F17"
    ],
    "keywords": [
      "functional strong law",
      "large numbers",
      "betti numbers",
      "components supporting topological cycles",
      "large connected components supporting"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 43,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}