{
  "id": "2010.13362",
  "version": "v1",
  "published": "2020-10-26T06:15:52.000Z",
  "updated": "2020-10-26T06:15:52.000Z",
  "title": "Quantitative two-scale stabilization on the Poisson space",
  "authors": [
    "Raphaël Lachièze-Rey",
    "Giovanni Peccati",
    "Xiaochuan Yang"
  ],
  "comment": "62 pages. Comments are welcome",
  "categories": [
    "math.PR"
  ],
  "abstract": "We establish inequalities for assessing the distance between the distribution of a (possibly multidimensional) functional of a Poisson random measure and that of a Gaussian element. Our bounds only involve add-one cost operators at the order one - that we evaluate and compare at two different scales - and are specifically tailored for studying the Gaussian fluctuations of sequences of geometric functionals displaying a form of weak stabilization - see Penrose and Yukich (2001) and Penrose (2005). Our main bounds extend the estimates recently exploited by Chatterjee and Sen (2017) in the proof of a quantitative version of the central limit theorem (CLT) for the length of the Poisson-based Euclidean minimal spanning tree (MST). We develop in full detail three applications of our bounds, namely: (i) to a quantitative multidimensional spatial CLT for functionals of the on-line nearest neighbor graph, (ii) to a quantitative multidimensional CLT involving functionals of the empirical measure associated with the edge-length of the Euclidean MST, and (iii) to a collection of multidimensional CLTs for geometric functionals of the excursion set of heavy-tailed shot noise random fields. Application (i) is based on a collection of general probabilistic approximations for strongly stabilizing functionals, that is of independent interest.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-10-26T06:15:52.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60F05",
      "60H07",
      "60G55",
      "60D05",
      "60G60"
    ],
    "keywords": [
      "quantitative two-scale stabilization",
      "poisson space",
      "euclidean minimal spanning tree",
      "multidimensional",
      "geometric functionals"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 62,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}