{
  "id": "1404.4776",
  "version": "v2",
  "published": "2014-04-18T12:55:25.000Z",
  "updated": "2015-11-28T09:55:04.000Z",
  "title": "Martingale inequalities of type Dzhaparidze and van Zanten",
  "authors": [
    "Xiequan Fan",
    "Ion Grama",
    "Quansheng Liu"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "Freedman's inequality is a supermartingale counterpart to Bennett's inequality. This result shows that the tail probabilities of a supermartingale is controlled by the quadratic characteristic and a uniform upper bound for the supermartingale difference sequence. Replacing the quadratic characteristic by $\\textrm{H}_k^y:= \\sum_{i=1}^k\\left(\\mathbf{E}(\\xi_i^2 |\\mathcal{F}_{i-1}) +\\xi_i^2\\textbf{1}_{\\{|\\xi_i|> y\\}}\\right),$ Dzhaparidze and van Zanten (\\emph{Stochastic Process. Appl.}, 2001) have extended Freedman's inequality to martingales with unbounded differences. In this paper, we prove that $\\textrm{H}_k^y$ can be refined to $\\textrm{G}_k^{y} :=\\sum_{i=1}^k \\left( \\mathbf{E}(\\xi_i^2\\textbf{1}_{\\{\\xi_i \\leq y\\}} |\\mathcal{F}_{i-1}) + \\xi_i^2\\textbf{1}_{\\{\\xi_i> y\\}}\\right).$ Moreover, we also establish two inequalities of type Dzhaparidze and van Zanten. These results extend Sason's inequality (\\emph{Statist. Probab. Lett.}, 2012) to the martingales with possibly unbounded differences and establish the connection between Sason's inequality and De la Pe\\~{n}a's inequality (\\emph{Ann.\\ Probab.,} 1999). An application to self-normalized deviations is given.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-04-18T12:55:25.000Z",
      "title": "Freedman's inequality with non-bounded martingale differences",
      "abstract": "Freedman's inequality is a martingale counterpart to Bernstein's inequality. This result shows that the tail probability of a martingale is controlled by the quadratic characteristic and a uniform upper bound for the martingale difference sequence. Replacing the quadratic characteristic with $\\textrm{H}_k^y:= \\sum_{i=1}^k\\left(\\mathbf{E}(\\xi_i^2 |\\mathcal{F}_{i-1}) +\\xi_i^2\\textbf{1}_{\\{|\\xi_i|> y\\}}\\right),$ Dzhaparidze and van Zanten (\\emph{Stochastic Process. Appl.}, 2001) have established a generalization of Freedman's inequality with non-bounded differences. In this paper, we refine $\\textrm{H}_k^y$ to $\\textrm{G}_k^{y} :=\\sum_{i=1}^k \\left(\\mathbf{E}(\\xi_i^2\\textbf{1}_{\\{\\xi_i \\leq y\\}} |\\mathcal{F}_{i-1}) + \\xi_i^2\\textbf{1}_{\\{\\xi_i> y\\}}\\right)$ with a different method based on changes of probability measure.",
      "comment": null,
      "journal": null,
      "doi": null,
      "authors": [
        "Xiequan Fan"
      ]
    },
    {
      "version": "v2",
      "updated": "2015-11-28T09:55:04.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60E15",
      "60F10",
      "60G42"
    ],
    "keywords": [
      "freedmans inequality",
      "non-bounded martingale differences",
      "quadratic characteristic",
      "uniform upper bound",
      "martingale difference sequence"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1404.4776F"
    }
  }
}