{
  "id": "1206.3967",
  "version": "v3",
  "published": "2012-06-18T15:26:07.000Z",
  "updated": "2014-09-08T08:55:39.000Z",
  "title": "Normal approximation of Poisson functionals in Kolmogorov distance",
  "authors": [
    "Matthias Schulte"
  ],
  "comment": "To appear in Journal of Theoretical Probability",
  "categories": [
    "math.PR"
  ],
  "abstract": "Peccati, Sole, Taqqu, and Utzet recently combined Stein's method and Malliavin calculus to obtain a bound for the Wasserstein distance of a Poisson functional and a Gaussian random variable. Convergence in the Wasserstein distance always implies convergence in the Kolmogorov distance at a possibly weaker rate. But there are many examples of central limit theorems having the same rate for both distances. The aim of this paper is to show this behaviour for a large class of Poisson functionals, namely so-called U-statistics of Poisson point processes. The technique used by Peccati et al. is modified to establish a similar bound for the Kolmogorov distance of a Poisson functional and a Gaussian random variable. This bound is evaluated for a U-statistic, and it is shown that the resulting expression is up to a constant the same as it is for the Wasserstein distance.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-06-06T09:31:33.000Z",
      "comment": null,
      "journal": null,
      "doi": null
    },
    {
      "version": "v3",
      "updated": "2014-09-08T08:55:39.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60F05",
      "60H07",
      "60G55"
    ],
    "keywords": [
      "poisson functional",
      "kolmogorov distance",
      "normal approximation",
      "wasserstein distance",
      "gaussian random variable"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1206.3967S"
    }
  }
}