{
  "id": "1806.07738",
  "version": "v1",
  "published": "2018-06-20T13:59:26.000Z",
  "updated": "2018-06-20T13:59:26.000Z",
  "title": "Free infinite divisibility for generalized power distributions with free Poisson term",
  "authors": [
    "Junki Morishita",
    "Yuki Ueda"
  ],
  "comment": "18 pages, 8 figures",
  "categories": [
    "math.PR"
  ],
  "abstract": "We study free infinite divisibility for the class of generalized power distributions with free Poisson term by using a complex analytic technique and a calculation for the free cumulants and Hankel determinants. In particular, our main result follows that (i) if $X$ follows the free Generalized Inverse Gaussian distribution, then $X^r$ follows the classes UI (Univalent Inverse Cauchy transform) and FR (Free Regular) when $|r|\\ge1$, (ii) if $S$ follows the standard semicircle law and $u\\ge 2$, then $(S+u)^r$ follows the classes UI and FR when $r\\le -1$, and (iii) if $B_p$ follows the beta distribution with parameters $p$ and $3/2$, then (a) $B_p^r$ follows the classes UI and FR when $|r|\\ge 1$ and $0<p\\le 1/2$, and (b) $B_p^r$ follows the classes UI and FR when $r\\le -1$ and $p>1/2$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-06-20T13:59:26.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46L54",
      "60E07",
      "32D15"
    ],
    "keywords": [
      "free infinite divisibility",
      "free poisson term",
      "generalized power distributions",
      "classes ui",
      "free generalized inverse gaussian distribution"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}