{
  "id": "1203.6763",
  "version": "v3",
  "published": "2012-03-30T10:30:35.000Z",
  "updated": "2014-11-26T13:25:22.000Z",
  "title": "Estimates for the concentration functions in the Littlewood--Offord problem",
  "authors": [
    "Yulia S. Eliseeva",
    "Friedrich Götze",
    "Andrei Yu. Zaitsev"
  ],
  "comment": "16 pages",
  "journal": "Zapiski Nauchnykh Seminarov POMI, 2013, vol. 420, p. 50-69",
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $X,X_1,...,X_n$ be independent identically distributed random variables. In this paper we study the behavior of the concentration functions of the weighted sums $\\sum\\limits_{k=1}^{n}a_k X_k$ with respect to the arithmetic structure of coefficients $a_k$. Such concentration results recently became important in connection with investigations about singular values of random matrices. In this paper we formulate and prove some refinements of a result of Vershynin (R. Vershynin, Invertibility of symmetric random matrices, arXiv:1102.0300. (2011). Published in Random Structures and Algorithms, v. 44, no. 2, 135--182 (2014)).",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-12-19T17:07:00.000Z",
      "abstract": "Let $X,X_1,\\ldots,X_n$ be independent identically distributed random variables. In this paper we study the behavior of the concentration functions of the weighted sums $\\sum\\limits_{k=1}^{n}a_k X_k$ with respect to the arithmetic structure of coefficients $a_k$. Such concentration results recently became important in connection with investigations about singular values of random matrices. In this paper we formulate and prove some refinements of a result of Vershynin (R. Vershynin, Invertibility of symmetric random matrices, arXiv:1102.0300. (2011). To appear in Random Structures and Algorithms).",
      "doi": null
    },
    {
      "version": "v3",
      "updated": "2014-11-26T13:25:22.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60F05",
      "60E15",
      "60G50"
    ],
    "keywords": [
      "concentration functions",
      "littlewood-offord problem",
      "independent identically distributed random variables",
      "symmetric random matrices",
      "random structures"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1203.6763E"
    }
  }
}