{
  "id": "2305.09547",
  "version": "v1",
  "published": "2023-05-16T15:42:43.000Z",
  "updated": "2023-05-16T15:42:43.000Z",
  "title": "Coherent distributions on the square $\\unicode{x2013}$ extreme points and asymptotics",
  "authors": [
    "Stanisław Cichomski",
    "Adam Osękowski"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $\\mathcal{C}$ denote the family of all coherent distributions on the unit square $[0,1]^2$, i.e. all those probability measures $\\mu$ for which there exists a random vector $(X,Y)\\sim \\mu$, a pair $(\\mathcal{G},\\mathcal{H})$ of $\\sigma$-fields and an event $E$ such that $X=\\mathbb{P}(E|\\mathcal{G})$, $Y=\\mathbb{P}(E|\\mathcal{H})$ almost surely. In this paper we examine the set $\\mathrm{ext}(\\mathcal{C})$ of extreme points of $\\mathcal{C}$ and provide its general characterisation. Moreover, we establish several structural properties of finitely-supported elements of $\\mathrm{ext}(\\mathcal{C})$. We apply these results to obtain the asymptotic sharp bound $$\\lim_{\\alpha \\to \\infty} \\alpha\\cdot \\Big(\\sup_{(X,Y)\\in \\mathcal{C}}\\mathbb{E}|X-Y|^{\\alpha}\\Big) = \\frac{2}{e}.$$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-05-16T15:42:43.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "extreme points",
      "coherent distributions",
      "asymptotic sharp bound",
      "unit square",
      "general characterisation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}