{
  "id": "2305.07885",
  "version": "v1",
  "published": "2023-05-13T10:10:56.000Z",
  "updated": "2023-05-13T10:10:56.000Z",
  "title": "Sharp deviation bounds and concentration phenomenon for the squared norm of a sub-gaussian vector",
  "authors": [
    "Vladimir Spokoiny"
  ],
  "comment": "arXiv admin note: text overlap with arXiv:2201.06327",
  "categories": [
    "math.PR"
  ],
  "abstract": "Let \\( \\Xv \\) be a Gaussian zero mean vector with \\( \\Var(\\Xv) = \\BBH \\). Then \\( \\| \\Xv \\|^{2} \\) well concentrates around its expectation \\( \\dimA = \\E \\| \\Xv \\|^{2} = \\tr \\BBH \\) provided that the latter is sufficiently large. Namely, \\( \\P\\bigl( \\| \\Xv \\|^{2} - \\tr \\BBH > 2 \\sqrt{\\xx \\tr(\\BBH^{2})} + 2 \\| \\BBH \\| \\xx \\bigr) \\leq \\ex^{-\\xx} \\) and \\( \\P\\bigl( \\| \\Xv \\|^{2} - \\tr \\BBH < - 2 \\sqrt{\\xx \\tr(\\BBH^{2})} \\bigr) \\leq \\ex^{-\\xx} \\); see \\cite{laurentmassart2000}. This note provides an extension of these bounds to the case of a sub-gaussian vector \\( \\Xv \\). The results are based on the recent advances in Laplace approximation from \\cite{SpLaplace2022}.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-05-13T10:10:56.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "62E15",
      "62E10"
    ],
    "keywords": [
      "sharp deviation bounds",
      "sub-gaussian vector",
      "concentration phenomenon",
      "squared norm",
      "gaussian zero mean vector"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}