{
  "id": "1007.4509",
  "version": "v2",
  "published": "2010-07-26T17:19:12.000Z",
  "updated": "2011-12-09T15:38:11.000Z",
  "title": "Fixed points of inhomogeneous smoothing transforms",
  "authors": [
    "Gerold Alsmeyer",
    "Matthias Meiners"
  ],
  "doi": "10.1080/10236198.2011.589514",
  "categories": [
    "math.PR"
  ],
  "abstract": "We consider the inhomogeneous version of the fixed-point equation of the smoothing transformation, that is, the equation $X \\stackrel{d}{=} C + \\sum_{i \\geq 1} T_i X_i$, where $\\stackrel{d}{=}$ means equality in distribution, $(C,T_1,T_2,...)$ is a given sequence of non-negative random variables and $X_1,X_2,...$ is a sequence of i.i.d.\\ copies of the non-negative random variable $X$ independent of $(C,T_1,T_2,...)$. In this situation, $X$ (or, more precisely, the distribution of $X$) is said to be a fixed point of the (inhomogeneous) smoothing transform. In the present paper, we give a necessary and sufficient condition for the existence of a fixed point. Further, we establish an explicit one-to-one correspondence with the solutions to the corresponding homogeneous equation with C=0. Using this correspondence, we present a full characterization of the set of fixed points under mild assumptions.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2011-12-09T15:38:11.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60E05",
      "60J80",
      "39B22"
    ],
    "keywords": [
      "fixed point",
      "inhomogeneous smoothing transforms",
      "non-negative random variable",
      "explicit one-to-one correspondence",
      "distribution"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1007.4509A"
    }
  }
}