{
  "id": "math/0608238",
  "version": "v1",
  "published": "2006-08-10T06:05:04.000Z",
  "updated": "2006-08-10T06:05:04.000Z",
  "title": "Coverage of space in Boolean models",
  "authors": [
    "Rahul Roy"
  ],
  "comment": "Published at http://dx.doi.org/10.1214/074921706000000158 in the IMS Lecture Notes--Monograph Series (http://www.imstat.org/publications/lecnotes.htm) by the Institute of Mathematical Statistics (http://www.imstat.org)",
  "journal": "IMS Lecture Notes--Monograph Series 2006, Vol. 48, 119-127",
  "doi": "10.1214/074921706000000158",
  "categories": [
    "math.CO",
    "math.PR"
  ],
  "abstract": "For a marked point process $\\{(x_i,S_i)_{i\\geq 1}\\}$ with $\\{x_i\\in \\Lambda:i\\geq 1\\}$ being a point process on $\\Lambda \\subseteq \\mathbb{R}^d$ and $\\{S_i\\subseteq R^d:i\\geq 1\\}$ being random sets consider the region $C=\\cup_{i\\geq 1}(x_i+S_i)$. This is the covered region obtained from the Boolean model $\\{(x_i+S_i):i\\geq 1\\}$. The Boolean model is said to be completely covered if $\\Lambda \\subseteq C$ almost surely. If $\\Lambda$ is an infinite set such that ${\\bf s}+\\Lambda \\subseteq \\Lambda$ for all ${\\bf s}\\in \\Lambda$ (e.g. the orthant), then the Boolean model is said to be eventually covered if ${\\bf t}+\\Lambda \\subseteq C$ for some ${\\bf t}$ almost surely. We discuss the issues of coverage when $\\Lambda$ is $\\mathbb{R}^d$ and when $\\Lambda$ is $[0,\\infty)^d$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2006-08-10T06:05:04.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C80",
      "05C40",
      "60K35"
    ],
    "keywords": [
      "boolean model",
      "infinite set",
      "random sets",
      "marked point process"
    ],
    "tags": [
      "monograph",
      "journal article",
      "lecture notes"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math......8238R"
    }
  }
}