{
  "id": "1501.07458",
  "version": "v1",
  "published": "2015-01-29T14:15:56.000Z",
  "updated": "2015-01-29T14:15:56.000Z",
  "title": "Convolution and convolution-root properties of long-tailed distributions",
  "authors": [
    "Hui Xu",
    "Sergey Foss",
    "Yuebao Wang"
  ],
  "comment": "22pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "We obtain a number of new general convolution properties related to long-tailed distributions. Then we show that the properties to be either long-tailed or generalised subexponential are not preserved under the convolution roots. Namely, we introduce two families of distributions such that each distribution is neither long-tailed nor generalised subexponential while its n-fold convolution is both long-tailed and generalised subexponential, for all n>=2. This leads to the negative answer to the conjecture of Embrechts and Goldie [10, 12] in the class of long-tailed and generalised subexponential distributions. Further, we analyse corresponding problems for distributions of random sums of random variables. In particular, we provide an example of a long-tailed and generalised subexponential infinitely divisible distribution whose Levy measure does not have these properties.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-01-29T14:15:56.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60E05",
      "60F10",
      "60G50"
    ],
    "keywords": [
      "long-tailed distributions",
      "convolution-root properties",
      "levy measure",
      "generalised subexponential distributions",
      "analyse corresponding problems"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150107458X"
    }
  }
}