{
  "id": "1903.03917",
  "version": "v1",
  "published": "2019-03-10T03:59:44.000Z",
  "updated": "2019-03-10T03:59:44.000Z",
  "title": "Iterates of Conditional Expectations: Convergence and Divergence",
  "authors": [
    "Guolie Lan",
    "Ze-Chun Hu",
    "Wei Sun"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "In this paper, we investigate the convergence and divergence of iterates of conditional expectation operators. We show that if $(\\Omega,\\cal{F},P)$ is a non-atomic probability space, then divergent iterates of conditional expectations involving 3 or 4 sub-$\\sigma$-fields of $\\cal{F}$ can be constructed for a large class of random variables in $L^2(\\Omega,\\cal{F},P)$. This settles in the negative a long-open conjecture. On the other hand, we show that if $(\\Omega,\\cal{F},P)$ is a purely atomic probability space, then iterates of conditional expectations involving any finite set of sub-$\\sigma$-fields of $\\cal{F}$ must converge for all random variables in $L^1(\\Omega,\\cal{F},P)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-03-10T03:59:44.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60A05",
      "60F15",
      "60F25"
    ],
    "keywords": [
      "divergence",
      "convergence",
      "random variables",
      "non-atomic probability space",
      "purely atomic probability space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}