{
  "id": "1404.4339",
  "version": "v1",
  "published": "2014-04-16T18:36:02.000Z",
  "updated": "2014-04-16T18:36:02.000Z",
  "title": "The Slide Dimension of Point Processes",
  "authors": [
    "Bill Ralph"
  ],
  "comment": "17 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "We associate with any finite subset of a metric space an infinite sequence of scale invariant numbers $\\rho_1,\\rho_2,\\dots$ derived from a variant of differential entropy called the genial entropy. As statistics for point processes, these numbers often appear to converge in simulations and we give examples where $1/\\rho_1$ converges to the Hausdorff dimension. We use the $\\rho_n$ to define a new notion of dimension called the slide dimension for a special class of point processes on metric spaces. The slide calculus is developed to define $\\rho_n$ and an explicit formula is derived for the calculation of $\\rho_1$. For a uniform random variable X on $[0,1]^n$, evidence is given that $\\rho_1(X) =1/n$ and $\\rho_2(X) =-\\pi^2/(6n^2)$ and simulations with a normal variable $Z$ suggest that $\\rho_1(Z) =4/\\pi$ and $\\rho_2(Z) =-1$. Some potential applications to spatial statistics are considered.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-04-16T18:36:02.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G55",
      "28A80"
    ],
    "keywords": [
      "point processes",
      "slide dimension",
      "metric space",
      "scale invariant numbers",
      "finite subset"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1404.4339R"
    }
  }
}