{
  "id": "1509.08614",
  "version": "v1",
  "published": "2015-09-29T07:18:32.000Z",
  "updated": "2015-09-29T07:18:32.000Z",
  "title": "Free infinite divisibility for powers of random variables",
  "authors": [
    "Takahiro Hasebe"
  ],
  "comment": "23 pages",
  "categories": [
    "math.PR",
    "math.OA"
  ],
  "abstract": "We prove that $X^r$ is free regular (stronger than freely infinitely divisible) if: (1) $X$ follows a free Poisson distribution without an atom at 0 and $r\\in(-\\infty,0]\\cup[1,\\infty)$; (2) $X$ follows a free Poisson distribution with an atom at 0 and $r\\geq1$; (3) $X$ follows a mixture of some HCM distributions and $|r|\\geq1$; (4) $X$ follows some beta distributions and $r$ is taken from some interval. In particular, if $S$ is a standard semicircular element then $|S|^r$ is freely infinitely divisible for $r\\in(-\\infty,0]\\cup[2,\\infty)$. Also we consider the symmetrization of the above probability measures, and in particular show that $|S|^r \\,{\\rm sign}(S)$ is freely infinitely divisible for $r\\geq2$. Therefore $S^n$ is freely infinitely divisible for every $n\\in\\mathbb{N}$. The results on free Poisson and semicircular random variables have a good correspondence with classical ID properties of powers of gamma and normal random variables.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-09-29T07:18:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46L54",
      "60E07"
    ],
    "keywords": [
      "free infinite divisibility",
      "freely infinitely divisible",
      "free poisson distribution",
      "semicircular random variables",
      "standard semicircular element"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150908614H"
    }
  }
}