{
  "id": "1706.06968",
  "version": "v1",
  "published": "2017-06-21T15:44:20.000Z",
  "updated": "2017-06-21T15:44:20.000Z",
  "title": "Exact Coupling of Random Walks on Polish Groups",
  "authors": [
    "James T. Murphy III"
  ],
  "comment": "14 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "Exact coupling of random walks is studied. Conditions for admitting a successful exact coupling are given that are necessary and in the Abelian case also sufficient. In the Abelian case, it is shown that a random walk $S$ with step-length distribution $\\mu$ started at $0$ admits a successful exact coupling with a version $S^x$ started at $x$ iff there is $n$ with $\\mu^{n} \\wedge \\mu^{n}(x+\\cdot) \\neq 0$. In particular, this paper solves a problem posed by H.~Thorisson on successful exact coupling of random walks on $\\mathbb{R}$. It is also noted that the set of such $x$ for which a successful exact coupling can be constructed is a Borel measurable group. Lastly, the weaker notion of possible exact coupling and its relationship to successful exact coupling is studied.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-06-21T15:44:20.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G50",
      "60F99",
      "28C10"
    ],
    "keywords": [
      "random walk",
      "successful exact coupling",
      "polish groups",
      "abelian case",
      "step-length distribution"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}