{
  "id": "1704.08333",
  "version": "v1",
  "published": "2017-04-26T20:03:51.000Z",
  "updated": "2017-04-26T20:03:51.000Z",
  "title": "Point-shifts of Point Processes on Topological Groups",
  "authors": [
    "James T. Murphy III"
  ],
  "comment": "26 pages (22 main + 4 appendix), 2 figures",
  "categories": [
    "math.PR"
  ],
  "abstract": "This paper focuses on covariant point-shifts of point processes on topological groups, which map points of a point process to other points of the point process in a translation invariant way. Foliations and connected components generated by point-shifts are studied, and the cardinality classification of connected components, previously known on Euclidean space, is generalized to unimodular groups. An explicit counterexample is also given on a non-unimodular group. Isomodularity of a point-shift is defined and identified as a key component in generalizations of Mecke's invariance theorem in the unimodular and non-unimodular cases. Isomodularity is also the deciding factor of when the reciprocal and reverse of a point-map corresponding to a bijective point-shift are equal in distribution. Finally, sufficient conditions for separating points of a point process are given.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-04-26T20:03:51.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37C85",
      "60G10",
      "60G55",
      "60G57",
      "28C10"
    ],
    "keywords": [
      "point process",
      "topological groups",
      "meckes invariance theorem",
      "connected components",
      "translation invariant way"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}