{
  "id": "1410.1050",
  "version": "v1",
  "published": "2014-10-04T14:39:39.000Z",
  "updated": "2014-10-04T14:39:39.000Z",
  "title": "Coupling on weighted branching trees",
  "authors": [
    "Ningyuan Chen",
    "Mariana Olvera-Cravioto"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "This paper considers linear functions constructed on two different weighted branching trees and provides explicit bounds for their Kantorovich-Rubinstein distance in terms of couplings of their corresponding generic branching vectors. By applying these bounds to a sequence of weighted branching trees, we derive the weak convergence of the corresponding linear processes. In the special case where sequence of trees converges to a weighted branching process, the limits can be represented as the endogenous solution to a stochastic fixed-point equation of the form $$R \\stackrel{D}{=} \\sum_{i=1}^N C_i R_i + Q,$$ where $(Q, N, \\{C_i\\})$ is a real-valued random vector with $N \\in \\mathbb{N} = \\{0, 1, 2, \\dots\\}$ and $\\{R_i\\}_{i \\in \\mathbb{N}}$ is a sequence of i.i.d. copies of $R$, independent of $(Q, N, \\{ C_i \\})$. The type of assumptions imposed on the generic branching vectors are suitable for applications in the analysis of a variety of random graph models.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-10-04T14:39:39.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J80",
      "60B10",
      "60H25"
    ],
    "keywords": [
      "weighted branching trees",
      "stochastic fixed-point equation",
      "random graph models",
      "explicit bounds",
      "kantorovich-rubinstein distance"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1410.1050C"
    }
  }
}