{
  "id": "1409.8457",
  "version": "v1",
  "published": "2014-09-30T09:53:03.000Z",
  "updated": "2014-09-30T09:53:03.000Z",
  "title": "A note on the Hanson-Wright inequality for random vectors with dependencies",
  "authors": [
    "Radosław Adamczak"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "We prove that quadratic forms in isotropic random vectors $X$ in $\\mathbb{R}^n$, possessing the convex concentration property with constant $K$, satisfy the Hanson-Wright inequality with constant $CK$, where $C$ is an absolute constant, thus eliminating the logarithmic (in the dimension) factors in a recent estimate by Vu and Wang. We also show that the concentration inequality for all Lipschitz functions implies a uniform version of the Hanson-Wright inequality for suprema of quadratic forms (in the spirit of the inequalities by Borell, Arcones-Gin\\'e and Ledoux-Talagrand). Previous results of this type relied on stronger isoperimetric properties of $X$ and in some cases provided an upper bound on the deviations rather than a concentration inequality. In the last part of the paper we show that the uniform version of the Hanson-Wright inequality for Gaussian vectors can be used to recover a recent concentration inequality for empirical estimators of the covariance operator of $B$-valued Gaussian variables due to Koltchinskii and Lounici.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-09-30T09:53:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60E15"
    ],
    "keywords": [
      "hanson-wright inequality",
      "concentration inequality",
      "dependencies",
      "quadratic forms",
      "uniform version"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1409.8457A"
    }
  }
}