{
  "id": "2011.08530",
  "version": "v1",
  "published": "2020-11-17T09:50:29.000Z",
  "updated": "2020-11-17T09:50:29.000Z",
  "title": "A Cramér--Wold device for infinite divisibility of $\\mathbb{Z}^d$-valued distributions",
  "authors": [
    "David Berger",
    "Alexander Lindner"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "We show that a Cram\\'er--Wold device holds for infinite divisibility of $\\mathbb{Z}^d$-valued distributions, i.e. that the distribution of a $\\mathbb{Z}^d$-valued random vector $X$ is infinitely divisible if and only if $\\mathcal{L}(a^T X)$ is infinitely divisible for all $a\\in \\mathbb{R}^d$, and that this in turn is equivalent to infinite divisibility of $\\mathcal{L}(a^T X)$ for all $a\\in \\mathbb{N}_0^d$. A key tool for proving this is a L\\'evy--Khintchine type representation with a signed L\\'evy measure for the characteristic function of a $\\mathbb{Z}^d$-valued distribution, provided the characteristic function is zero-free.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-11-17T09:50:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60E07"
    ],
    "keywords": [
      "infinite divisibility",
      "valued distribution",
      "cramér-wold device",
      "characteristic function",
      "cramer-wold device holds"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}