{
  "id": "0907.2693",
  "version": "v2",
  "published": "2009-07-15T20:06:38.000Z",
  "updated": "2009-10-20T12:22:17.000Z",
  "title": "A CLT for the third integrated moment of Brownian local time increments",
  "authors": [
    "Jay Rosen"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $\\{L^{x}_{t} ; (x,t)\\in R^{1}\\times R^{1}_{+}\\}$ denote the local time of Brownian motion. Our main result is to show that for each fixed $t$ $${\\int (L^{x+h}_t- L^x_t)^3 dx-12h\\int (L^{x+h}_t - L^x_t)L^x_t dx-24h^{2}t\\over h^2} \\stackrel{\\mathcal{L}}{\\Longrightarrow}\\sqrt{192}(\\int (L^x_t)^3dx)^{1/2}\\eta$$ as $h\\to 0$, where $\\eta$ is a normal random variable with mean zero and variance one that is independent of $L^{x}_{t}$. This generalizes our previous result for the second moment. We also explain why our approach will not work for higher moments",
  "revisions": [
    {
      "version": "v2",
      "updated": "2009-10-20T12:22:17.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60F05",
      "60J55",
      "60J65"
    ],
    "keywords": [
      "brownian local time increments",
      "third integrated moment",
      "second moment",
      "brownian motion",
      "mean zero"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0907.2693R"
    }
  }
}