{
  "id": "1309.6940",
  "version": "v1",
  "published": "2013-09-26T15:34:31.000Z",
  "updated": "2013-09-26T15:34:31.000Z",
  "title": "Strong representation of weak convergence",
  "authors": [
    "Zhidong Bai",
    "Jiang Hu"
  ],
  "comment": "11 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "Skorokhod's representation theorem states that if on a Polish space, there is defined a weakly convergent sequence of probability measures $\\mu_n\\stackrel{w}\\to\\mu_0,$ as $n\\to \\infty$, then there exist a probability space $(\\Omega, \\mathscr F, P)$ and a sequence of random elements $X_n$ such that $X_n\\to X$ almost surely and $X_n$ has the distribution function $\\mu_n$, $n=0,1,2,\\cdots$. In this paper, we shall extend the Skorokhod representation theorem to the case where if there are a sequence of separable metric spaces $S_n$, a sequence of probability measures $\\mu_n$ and a sequence of measurable mappings $\\varphi_n$ such that $\\mu_n\\varphi_n^{-1}\\stackrel {w}\\to\\mu_0$, then there exist a probability space $(\\Omega,\\mathscr F,P)$ and $S_n$-valued random elements $X_n$ defined on $\\Omega$, with distribution $\\mu_n$ and such that $\\varphi_n(X_n)\\to X_0$ almost surely. In addition, we present several applications of our result including some results in random matrix theory, while the original Skorokhod representation theorem is not applicable.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-09-26T15:34:31.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "weak convergence",
      "strong representation",
      "skorokhods representation theorem states",
      "original skorokhod representation theorem",
      "random elements"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1309.6940B"
    }
  }
}