{
  "id": "1909.05584",
  "version": "v1",
  "published": "2019-09-12T11:50:45.000Z",
  "updated": "2019-09-12T11:50:45.000Z",
  "title": "Large deviation inequalities for martingales in Banach spaces",
  "authors": [
    "Xiequan Fan",
    "Davide Giraudo"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $(X_i, \\mathcal{F}_i)_{i\\geq1}$ be a martingale difference sequence in a smooth Banach space. Let $S_n=\\sum_{i=1}^nX_i, n\\geq 1,$ be the partial sums of $(X_i, \\mathcal{F}_i)_{i\\geq 1}$. We give upper bounds on the quantity $\\mathbb{P}\\left(\\max_{1\\leq k\\leq n}\\lVert S_k\\rVert>nx\\right)$ in terms of $ n\\geq 1$ and $x>0$ in two different situations: when the martingale differences have uniformly bounded exponential moments and when the decay of the tail of the increments is polynomial.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-09-12T11:50:45.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60F10",
      "60G10",
      "60E15"
    ],
    "keywords": [
      "large deviation inequalities",
      "martingale difference sequence",
      "smooth banach space",
      "partial sums",
      "uniformly bounded exponential moments"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}