{
  "id": "0912.2795",
  "version": "v2",
  "published": "2009-12-15T20:46:08.000Z",
  "updated": "2009-12-15T21:18:49.000Z",
  "title": "An improvement of the Berry--Esseen inequality with applications to Poisson and mixed Poisson random sums",
  "authors": [
    "Victor Korolev",
    "Irina Shevtsova"
  ],
  "comment": "33 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "By a modification of the method that was applied in (Korolev and Shevtsova, 2009), here the inequalities $$\\rho(F_n,\\Phi)\\le\\frac{0.335789(\\beta^3+0.425)}{\\sqrt{n}}$$ and $$\\rho(F_n,\\Phi)\\le \\frac{0.3051(\\beta^3+1)}{\\sqrt{n}} $$ are proved for the uniform distance $\\rho(F_n,\\Phi)$ between the standard normal distribution function $\\Phi$ and the distribution function $F_n$ of the normalized sum of an arbitrary number $n\\ge1$ of independent identically distributed random variables with zero mean, unit variance and finite third absolute moment $\\beta^3$. The first of these inequalities sharpens the best known version of the classical Berry--Esseen inequality since $0.335789(\\beta^3+0.425)\\le0.335789(1+0.425)\\beta^3<0.4785\\beta^3$ by virtue of the condition $\\beta^3\\ge1$, and 0.4785 is the best known upper estimate of the absolute constant in the classical Berry--Esseen inequality. The second inequality is applied to lowering the upper estimate of the absolute constant in the analog of the Berry--Esseen inequality for Poisson random sums to 0.3051 which is strictly less than the least possible value of the absolute constant in the classical Berry--Esseen inequality. As a corollary, the estimates of the rate of convergence in limit theorems for compound mixed Poisson distributions are refined.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2009-12-15T21:18:49.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60F05"
    ],
    "keywords": [
      "mixed poisson random sums",
      "classical berry-esseen inequality",
      "absolute constant",
      "identically distributed random variables",
      "improvement"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 33,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0912.2795K"
    }
  }
}