{
  "id": "1810.09681",
  "version": "v1",
  "published": "2018-10-23T06:44:29.000Z",
  "updated": "2018-10-23T06:44:29.000Z",
  "title": "On a bound of the absolute constant in the Berry--Esseen inequality for i.i.d. Bernoulli random variables",
  "authors": [
    "Anatolii Zolotukhin",
    "Sergei Nagaev",
    "Vladimir Chebotarev"
  ],
  "comment": "Published at https://doi.org/10.15559/18-VMSTA113 in the Modern Stochastics: Theory and Applications (https://www.i-journals.org/vtxpp/VMSTA) by VTeX (http://www.vtex.lt/)",
  "journal": "Modern Stochastics: Theory and Applications 2018, Vol. 5, No. 3, 385-410",
  "doi": "10.15559/18-VMSTA113",
  "categories": [
    "math.PR"
  ],
  "abstract": "It is shown that the absolute constant in the Berry--Esseen inequality for i.i.d. Bernoulli random variables is strictly less than the Esseen constant, if $1\\le n\\le 500000$, where $n$ is a number of summands. This result is got both with the help of a supercomputer and an interpolation theorem, which is proved in the paper as well. In addition, applying the method developed by S. Nagaev and V. Chebotarev in 2009--2011, an upper bound is obtained for the absolute constant in the Berry--Esseen inequality in the case under consideration, which differs from the Esseen constant by no more than 0.06%. As an auxiliary result, we prove a bound in the local Moivre--Laplace theorem which has a simple and explicit form. Despite the best possible result, obtained by J. Schulz in 2016, we propose our approach to the problem of finding the absolute constant in the Berry--Esseen inequality for two-point distributions since this approach, combining analytical methods and the use of computers, could be useful in solving other mathematical problems.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-10-23T06:44:29.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "bernoulli random variables",
      "berry-esseen inequality",
      "absolute constant",
      "esseen constant",
      "local moivre-laplace theorem"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}