{
  "id": "0806.3188",
  "version": "v1",
  "published": "2008-06-19T13:45:00.000Z",
  "updated": "2008-06-19T13:45:00.000Z",
  "title": "Existence of a critical point for the infinite divisibility of squares of Gaussian vectors in $R^{2}$ with non--zero mean",
  "authors": [
    "Michael B. Marcus",
    "Jay Rosen"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $G=(G_{1},G_{2})$ be a Gaussian vector in $R^{2}$ with $EG_{1}G_{2}\\neq 0$. Let $c_{1},c_{2}\\in R^{1}$. A necessary and sufficient condition for $G=((G_{1}+c_{1}\\alpha)^{2},(G_{2}+c_{2}\\alpha)^{2})$ to be infinitely divisible for all $\\alpha\\in R^{1}$ is that \\[ \\Ga_{i,i}\\geq \\frac{c_{i}}{c_{j}}\\Ga_{i,j}>0\\qquad\\forall 1\\le i\\ne j\\le 2.\\] In this paper we show that when this does not hold there exists an $0<\\alpha_{0}<\\ff $ such that $G=((G_{1}+c_{1}\\alpha)^{2},(G_{2}+c_{2}\\alpha)^{2})$ is infinitely divisible for all $|\\alpha|\\leq \\alpha_{0}$ but not for any $|\\al|>\\al_{0}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2008-06-19T13:45:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G15",
      "60E07"
    ],
    "keywords": [
      "gaussian vector",
      "non-zero mean",
      "infinite divisibility",
      "critical point",
      "sufficient condition"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0806.3188M"
    }
  }
}