{
  "id": "1405.6505",
  "version": "v2",
  "published": "2014-05-26T08:51:24.000Z",
  "updated": "2015-02-09T08:54:45.000Z",
  "title": "Precise Large Deviation Results for Products of Random Matrices",
  "authors": [
    "Dariusz Buraczewski",
    "Sebastian Mentemeier"
  ],
  "comment": "39 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "The theorem of Furstenberg and Kesten provides a strong law of large numbers for the norm of a product of random matrices. This can be extended under various assumptions, covering nonnegative as well as invertible matrices, to a law of large numbers for the norm of a vector on which the matrices act. We prove corresponding precise large deviation results, generalizing the Bahadur-Rao theorem to this situation. Therefore, we obtain a third-order Edgeworth expansion for the cumulative distribution function of the vector norm. This result in turn relies on an application of the Nagaev-Guivarch method. Our result is then used to study matrix recursions, arising e.g. in financial time series, and to provide precise large deviation estimates there.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-05-26T08:51:24.000Z",
      "comment": "32 pages",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2015-02-09T08:54:45.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60F10",
      "60H25"
    ],
    "keywords": [
      "random matrices",
      "precise large deviation estimates",
      "large numbers",
      "corresponding precise large deviation results",
      "study matrix recursions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 39,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1405.6505B"
    }
  }
}