{
  "id": "math/0201273",
  "version": "v1",
  "published": "2002-01-29T09:37:02.000Z",
  "updated": "2002-01-29T09:37:02.000Z",
  "title": "Entropy and a generalisation of `Poincare's Observation'",
  "authors": [
    "Oliver Johnson"
  ],
  "comment": "12 pages",
  "journal": "Mathematical Proceedings of the Cambridge Philosophical Society, Vol 135/2, 2003, pages 375-384",
  "doi": "10.1017/S0305004103006881",
  "categories": [
    "math.PR",
    "math.ST",
    "stat.TH"
  ],
  "abstract": "Consider a sphere of radius root(n) in n dimensions, and consider X, a random variable uniformly distributed on its surface. Poincare's Observation states that for large n, the distribution of the first k coordinates of X is close in total variation distance to the standard normal N(0,I_k). In this paper, we consider a larger family of manifolds, and X taking a more general distribution on the surfaces. We establish a bound in the stronger Kullback--Leibler sense of relative entropy, and discuss its sharpness, providing a necessary condition for convergence in this sense. We show how our results imply the equivalence of ensembles for a wider class of test functions than is standard. We also deduce results of de Finetti type, concerning a generalisation of the idea of orthogonal invariance.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2002-01-29T09:37:02.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60F99",
      "62B10",
      "94A17"
    ],
    "keywords": [
      "generalisation",
      "poincares observation states",
      "total variation distance",
      "stronger kullback-leibler sense",
      "radius root"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "journal": "Mathematical Proceedings of the Cambridge Philosophical Society",
      "year": 2003,
      "month": "Sep",
      "volume": 135,
      "number": 2,
      "pages": 375
    },
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2003MPCPS.135..375J"
    }
  }
}