{
  "id": "1803.05150",
  "version": "v1",
  "published": "2018-03-14T07:09:22.000Z",
  "updated": "2018-03-14T07:09:22.000Z",
  "title": "Bernstein type inequalities for self-normalized martingales with applications",
  "authors": [
    "Xiequan Fan",
    "Shen Wang"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "For self-normalized martingales with conditionally symmetric differences, de la Pe\\~{n}a [A general class of exponential inequalities for martingales and ratios. Ann. Probab. 27, No.1, 537-564] established the Gaussian type exponential inequalities. Bercu and Touati [Exponential inequalities for self-normalized martingales with applications. Ann. Appl. Probab. 18: 1848-1869] extended de la Pe\\~{n}a's inequalities to martingales with differences heavy on left. In this paper, we establish Bernstein type exponential inequalities for self-normalized martingales with differences bounded from below. Moreover, applications to self-normalized sums, t-statistics and autoregressive processes are discussed.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-03-14T07:09:22.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G42",
      "60E15",
      "60F10"
    ],
    "keywords": [
      "self-normalized martingales",
      "bernstein type inequalities",
      "applications",
      "gaussian type exponential inequalities",
      "establish bernstein type exponential inequalities"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}