{
  "id": "0712.3440",
  "version": "v1",
  "published": "2007-12-20T14:48:54.000Z",
  "updated": "2007-12-20T14:48:54.000Z",
  "title": "Domains of attraction of the random vector $(X,X^2)$ and applications",
  "authors": [
    "Edward Omey"
  ],
  "comment": "To appear in a Special Volume of Stochastics: An International Journal of Probability and Stochastic Processes (http://www.informaworld.com/openurl?genre=journal%26issn=1744-2508) edited by N.H. Bingham and I.V. Evstigneev which will be reprinted as Volume 57 of the IMS Lecture Notes Monograph Series (http://imstat.org/publications/lecnotes.htm)",
  "categories": [
    "math.PR"
  ],
  "abstract": "Many statistics are based on functions of sample moments. Important examples are the sample variance $s_{n-1}^2$, the sample coefficient of variation SV(n), the sample dispersion SD(n) and the non-central $t$-statistic $t(n)$. The definition of these quantities makes clear that the vector defined by (\\sum_{i=1}^nX_i,\\sum_{i=1}^nX_i^2) plays an important role. In studying the asymptotic behaviour of this vector we start by formulating best possible conditions under which the vector $(X,X^2)$ belongs to a bivariate domain of attraction of a stable law. This approach is new, uniform and simple. Our main results include a full discussion of the asymptotic behaviour of SV(n), SD(n) and $t^2(n)$. For simplicity, in restrict ourselves to positive random variables $X$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2007-12-20T14:48:54.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60E05",
      "60F05",
      "62E20",
      "91B70"
    ],
    "keywords": [
      "random vector",
      "attraction",
      "applications",
      "asymptotic behaviour",
      "sample dispersion sd"
    ],
    "tags": [
      "monograph",
      "lecture notes"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0712.3440O"
    }
  }
}