{
  "id": "1510.06451",
  "version": "v1",
  "published": "2015-10-21T22:48:13.000Z",
  "updated": "2015-10-21T22:48:13.000Z",
  "title": "Thin tails of fixed points of the nonhomogeneous smoothing transform",
  "authors": [
    "Gerold Alsmeyer",
    "Piotr Dyszewski"
  ],
  "comment": "26 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "For a given random sequence $(C,T_{1},T_{2},\\ldots)$ with nonzero $C$ and a.s. finite number of nonzero $T_{k}$, the nonhomogeneous smoothing transform $\\mathcal{S}$ maps the law of a real random variable $X$ to the law of $\\sum_{k\\ge 1}T_{k}X_{k}+C$, where $X_{1},X_{2},\\ldots$ are independent copies of $X$ and also independent of $(C,T_{1},T_{2},\\ldots)$. This law is a fixed point of $\\mathcal{S}$ if the stochastic fixed-point equation (SFPE) $X\\stackrel{d}{=}\\sum_{k\\ge 1}T_{k}X_{k}+C$ holds true, where $\\stackrel{d}{=}$ denotes equality in law. Under suitable conditions including $\\mathbb{E} C=0$, $\\mathcal{S}$ possesses a unique fixed point within the class of centered distributions, called the canonical solution to the above SFPE because it can be obtained as a certain martingale limit in an associated weighted branching model. The present work provides conditions on $(C,T_{1},T_{2},\\ldots)$ such that the canonical solution exhibits right and/or left Poisson tails and the abscissa of convergence of its moment generating function can be determined. As a particular application, the right tail behavior of the Quicksort distribution is found.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-10-21T22:48:13.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60H25",
      "60E10"
    ],
    "keywords": [
      "nonhomogeneous smoothing transform",
      "fixed point",
      "thin tails",
      "left poisson tails",
      "stochastic fixed-point equation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}