{
  "id": "1402.4147",
  "version": "v1",
  "published": "2014-02-17T21:12:41.000Z",
  "updated": "2014-02-17T21:12:41.000Z",
  "title": "Fixed points of multivariate smoothing transforms with scalar weights",
  "authors": [
    "Alexander Iksanov",
    "Matthias Meiners"
  ],
  "comment": "43 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "Given a sequence $(C_1,\\ldots,C_d,T_1,T_2,\\ldots)$ of real-valued random variables with $N := \\#\\{j \\geq 1: T_j \\not = 0\\} < \\infty$ almost surely, there is an associated smoothing transformation which maps a distribution $P$ on $\\mathbb{R}^d$ to the distribution of $\\sum_{j \\geq 1} T_j \\mathbf{X}^{(j)} + \\mathbf{C}$ where $\\mathbf{C} = (C_1,\\ldots,C_d)$ and $(\\mathbf{X}^{(j)})_{j \\geq 1}$ is a sequence of independent random vectors with distribution $P$ independent of $(C_1,\\ldots,C_d,T_1,T_2,\\ldots)$. We are interested in the fixed points of this mapping. By improving on the techniques developed in [G. Alsmeyer, J.D. Biggins, and M. Meiners. The functional equation of the smoothing transform {\\em Ann. Probab.}, 40(5):2069--2105, 2012] and [G. Alsmeyer and M. Meiners. Fixed points of the smoothing transform: two-sided solutions. {\\em Probab. Theory Related Fields}, 155(1-2):165--199, 2013], we determine the set of all fixed points under weak assumptions on $(C_1,\\ldots,C_d,T_1,T_2,\\ldots)$. In contrast to earlier studies, this includes the most intricate case when the $T_j$ take both positive and negative values with positive probability. In this case, in some situations, the set of fixed points is a subset of the corresponding set when the $T_j$ are replaced by their absolute values, while in other situations, additional solutions arise.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-02-17T21:12:41.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J80",
      "39B32"
    ],
    "keywords": [
      "fixed points",
      "multivariate smoothing transforms",
      "scalar weights",
      "distribution",
      "independent random vectors"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 43,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1402.4147I"
    }
  }
}