{
  "id": "0906.3037",
  "version": "v1",
  "published": "2009-06-16T22:33:12.000Z",
  "updated": "2009-06-16T22:33:12.000Z",
  "title": "On the density of the sum of two independent Student t-random vectors",
  "authors": [
    "C. Berg",
    "C. Vignat"
  ],
  "comment": "1 figure",
  "categories": [
    "math.PR"
  ],
  "abstract": "In this paper, we find an expression for the density of the sum of two independent $d-$dimensional Student $t-$random vectors $\\mathbf{X}$ and $\\mathbf{Y}$ with arbitrary degrees of freedom. As a byproduct we also obtain an expression for the density of the sum $\\mathbf{N}+\\mathbf{X}$, where $\\mathbf{N}$ is normal and $\\mathbf{X}$ is an independent Student $t-$vector. In both cases the density is given as an infinite series \\[ \\sum_{n=0}^{\\infty} c_{n}f_{n} \\] where $f_{n}$ is a sequence of probability densities on $\\mathbb{R}^{d}$ and $(c_{n} )$ is a sequence of positive numbers of sum 1, i.e. the distribution of a non-negative integer-valued random variable $C$, which turns out to be infinitely divisible for $d=1$ and $d=2.$ When $d=1$ and the degrees of freedom of the Student variables are equal, we recover an old result of Ruben.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-06-16T22:33:12.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60E05"
    ],
    "keywords": [
      "independent student t-random vectors",
      "old result",
      "arbitrary degrees",
      "expression",
      "dimensional student"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0906.3037B"
    }
  }
}