{
  "id": "2210.08653",
  "version": "v1",
  "published": "2022-10-16T22:40:30.000Z",
  "updated": "2022-10-16T22:40:30.000Z",
  "title": "A note on positive association",
  "authors": [
    "Jeff Kahn"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "We show that if ${\\mathcal A},{\\mathcal B},{\\mathcal C}$ are increasing subsets of $\\Omega:=\\{0,1\\}^n$ with ${\\mathcal A}\\neq\\emptyset$, then with respect to any product probability measure on $\\Omega$, \\[ \\mbox{if each of the pairs $\\{{\\mathcal A}\\cap{\\mathcal B},{\\mathcal C}\\}$, $\\{{\\mathcal A}\\cap {\\mathcal C},{\\mathcal A}\\}$ is independent, then ${\\mathcal B}$ and ${\\mathcal C}$ are independent.} \\] This implies an answer to a motivating question of J. Steif, and is related to a basic, still open variant of that question, and to a well-known conjecture of S. Sahi.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-10-16T22:40:30.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60C05"
    ],
    "keywords": [
      "positive association",
      "product probability measure",
      "well-known conjecture",
      "open variant",
      "independent"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}