{
  "id": "1206.4807",
  "version": "v1",
  "published": "2012-06-21T08:23:55.000Z",
  "updated": "2012-06-21T08:23:55.000Z",
  "title": "Distances between Poisson k-flats",
  "authors": [
    "Matthias Schulte",
    "Christoph Thaele"
  ],
  "journal": "Methodol. Comput. Appl. Probab. 16, 311-329 (2014)",
  "categories": [
    "math.PR",
    "math.MG"
  ],
  "abstract": "The distances between flats of a Poisson $k$-flat process in the $d$-dimensional Euclidean space with $k<d/2$ are discussed. Continuing an approach originally due to Rolf Schneider, the number of pairs of flats having distance less than a given threshold and midpoint in a fixed compact and convex set is considered. For a family of increasing convex subsets, the asymptotic variance is computed and a central limit theorem with an explicit rate of convergence is proven. Moreover, the asymptotic distribution of the $m$-th smallest distance between two flats is investigated and it is shown that the ordered distances form asymptotically after suitable rescaling an inhomogeneous Poisson point process on the positive real axis. A similar result with a homogeneous limiting process is derived for distances around a fixed, strictly positive value. Our proofs rely on recent findings based on the Wiener-It\\^o chaos decomposition and the Malliavin-Stein method.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-06-21T08:23:55.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60D05",
      "60F05",
      "60G55",
      "60H07"
    ],
    "keywords": [
      "poisson k-flats",
      "dimensional euclidean space",
      "th smallest distance",
      "inhomogeneous poisson point process",
      "central limit theorem"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1206.4807S"
    }
  }
}