{
  "id": "1601.00078",
  "version": "v1",
  "published": "2016-01-01T12:49:26.000Z",
  "updated": "2016-01-01T12:49:26.000Z",
  "title": "A Characterization of the Normal Distribution by the Independence of a Pair of Random Vectors",
  "authors": [
    "Wiktor Ejsmont"
  ],
  "comment": "6 pages",
  "categories": [
    "math.PR",
    "math.ST",
    "stat.TH"
  ],
  "abstract": "Kagan and Shalaevski 1967 have shown that if the random variables $X_1,\\dots,X_n$ are independent and identically distributed and the distribution of $\\sum_{i=1}^n(X_i+a_i)^2$ $a_i\\in \\mathbb{R}$ depends only on $\\sum_{i=1}^na_i^2$ , then each $X_i$ follows the normal distribution $N(0, \\sigma)$. Cook 1971 generalized this result replacing independence of all $X_i$ by the independence of $(X_1,\\dots, X_m) \\textrm{ and } (X_{m+1},\\dots,X_n )$ and removing the requirement that $X_i$ have the same distribution. In this paper, we will give other characterizations of the normal distribution which are formulated in a similar spirit.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-01-01T12:49:26.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "normal distribution",
      "random vectors",
      "characterization",
      "similar spirit",
      "result replacing independence"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 6,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160100078E"
    }
  }
}