{
  "id": "1111.6554",
  "version": "v1",
  "published": "2011-11-28T19:18:37.000Z",
  "updated": "2011-11-28T19:18:37.000Z",
  "title": "On the absolute constants in the Berry-Esseen type inequalities for identically distributed summands",
  "authors": [
    "Irina Shevtsova"
  ],
  "comment": "7 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "By a modification of the method that was applied in (Korolev and Shevtsova, 2010), here the inequalities $\\Delta_n\\leq0.3328(\\beta_3+0.429)/\\sqrt{n}$ and $\\Delta_n\\leq0.33554(\\beta_3+0.415)/\\sqrt{n}$ are proved for the uniform distance $\\Delta_n$ between the standard normal distribution function and the distribution function of the normalized sum of an arbitrary number $n\\geq1$ of independent identically distributed random variables with zero mean, unit variance and finite third absolute moment $\\beta_3$. The first of these two inequalities improves one that was proved in (Korolev and Shevtsova, 2010), and as well sharpens the best known upper estimate for the absolute constant $C_0$ in the classical Berry--Esseen inequality to be $C_0<0.4756$, since $0.3328(\\beta_3+0.429)\\leq0.3328\\cdot1.429\\beta_3<0.4756\\beta_3$ by virtue of the condition $\\beta_3\\geq1$. The second of these inequalities is also a structural improvement of the classical Berry--Esseen inequality, and as well sharpens the upper estimate for $C_0$ still more to be $C_0<0.4748$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-11-28T19:18:37.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60F05"
    ],
    "keywords": [
      "berry-esseen type inequalities",
      "absolute constant",
      "identically distributed summands",
      "identically distributed random variables",
      "classical berry-esseen inequality"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 7,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1111.6554S"
    }
  }
}