{
  "id": "1111.6358",
  "version": "v1",
  "published": "2011-11-28T07:25:54.000Z",
  "updated": "2011-11-28T07:25:54.000Z",
  "title": "Bounds for tail probabilities of martingales using skewness and kurtosis",
  "authors": [
    "Vidmantas Bentkus",
    "Tomas Juškevičius"
  ],
  "comment": "Lithuanian Mathematical Journal (2008)",
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $M_n= \\fsu X1n$ be a sum of independent random variables such that $ X_k\\leq 1$, $\\E X_k =0$ and $\\E X_k^2=\\s_k^2$ for all $k$. Hoeffding 1963, Theorem 3, proved that $$\\P{M_n \\geq nt}\\leq H^n(t,p),\\quad H(t,p)= \\bgl(1+qt/p\\bgr)^{p +qt} \\bgl({1-t}\\bgr)^{q -qt}$$ with $$q=\\ffrac 1{1+\\s^2},\\quad p=1-q, \\quad \\s^2 =\\ffrac {\\s_1^2+...+\\s_n^2}n,\\quad 0<t<1.$$ Bentkus 2004 improved Hoeffding's inequalities using binomial tails as upper bounds. Let $\\ga_k =\\E X_k^3/\\s_k^3$ and $ \\vk_k= \\E X_k^4/\\s_k^4$ stand for the skewness and kurtosis of $X_k$. In this paper we prove (improved) counterparts of the Hoeffding inequality replacing $\\s^2$ by certain functions of $\\fs \\ga 1n$ respectively $\\fs \\vk 1n$. Our bounds extend to a general setting where $X_k$ are martingale differences, and they can combine the knowledge of skewness and/or kurtosis and/or variances of ~$X_k$. Up to factors bounded by $e^2/2$ the bounds are final. All our results are new since no inequalities incorporating skewness or kurtosis control so far are known.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-11-28T07:25:54.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "tail probabilities",
      "independent random variables",
      "upper bounds",
      "kurtosis control",
      "bounds extend"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1111.6358B"
    }
  }
}