{
  "id": "2007.12241",
  "version": "v1",
  "published": "2020-07-23T20:26:58.000Z",
  "updated": "2020-07-23T20:26:58.000Z",
  "title": "The Heyde theorem on a group $\\mathbb{R}^n\\times D$, where $D$ is a discrete Abelian group",
  "authors": [
    "Margaryta Myronyuk"
  ],
  "categories": [
    "math.FA",
    "math.ST",
    "stat.TH"
  ],
  "abstract": "Heyde proved that a Gaussian distribution on the real line is characterized by the symmetry of the conditional distribution of one linear statistic given another. The present article is devoted to a group analogue of the Heyde theorem. We describe distributions of independent random variables $\\xi_1$, $\\xi_2$ with values in a group $X=\\mathbb{R}^n\\times D$, where $D$ is a discrete Abelian group, which are characterized by the symmetry of the conditional distribution of the linear statistic $L_2 = \\xi_1 + \\delta\\xi_2$ given $L_1 = \\xi_1 + \\xi_2$, where $\\delta$ is a topological automorphism of $X$ such that ${Ker}(I+\\delta)=\\{0\\}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-07-23T20:26:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60B15",
      "62E10"
    ],
    "keywords": [
      "discrete abelian group",
      "heyde theorem",
      "conditional distribution",
      "linear statistic",
      "independent random variables"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}