{
  "id": "1509.09261",
  "version": "v1",
  "published": "2015-09-30T17:20:58.000Z",
  "updated": "2015-09-30T17:20:58.000Z",
  "title": "Polar decomposition of scale-homogeneous measures with application to Lévy measures of strictly stable laws",
  "authors": [
    "Steven N. Evans",
    "Ilya Molchanov"
  ],
  "comment": "22 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "A scaling on some space is a measurable action of the group of positive real numbers. A measure on a measurable space equipped with a scaling is said to be $\\alpha$-homogeneous for some nonzero real number $\\alpha$ if the mass of any measurable set scaled by any factor $t > 0$ is the multiple $t^{-\\alpha}$ of the set's original mass. It is shown rather generally that given an $\\alpha$-homogeneous measure on a measurable space there is a measurable bijection between the space and the Cartesian product of a subset of the space and the positive real numbers (that is, a \"system of polar coordinates\") such that the push-forward of the $\\alpha$-homogeneous measure by this bijection is the product of a probability measure on the first component (that is, on the \"angular\" component) and an $\\alpha$-homogeneous measure on the positive half-line (that is, on the \"radial\" component). This result is applied to the intensity measures of Poisson processes that arise in L\\'evy-Khinchin-It\\^o-like representations of infinitely divisible random elements. It is established that if a strictly stable random element in a convex cone admits a series representation as the sum of points of a Poisson process, then it necessarily has a LePage representation as the sum of i.i.d. random elements of the cone scaled by the successive points of an independent unit intensity Poisson process on the positive half-line each raised to the power $-\\frac{1}{\\alpha}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-09-30T17:20:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28A50",
      "28C10",
      "60B15",
      "60E07"
    ],
    "keywords": [
      "strictly stable laws",
      "lévy measures",
      "polar decomposition",
      "scale-homogeneous measures",
      "random element"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}