{
  "id": "0901.1102",
  "version": "v1",
  "published": "2009-01-08T17:55:28.000Z",
  "updated": "2009-01-08T17:55:28.000Z",
  "title": "A CLT for the L^{2} modulus of continuity of Brownian local time",
  "authors": [
    "Xia Chen",
    "Wenbo Li",
    "Michael B. Marcus",
    "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 and \\[ \\alpha_{t}:=\\int_{-\\infty}^{\\infty} (L^{x}_{t})^{2} dx . \\] Let $\\eta=N(0,1)$ be independent of $\\alpha_{t}$. For each fixed $t$ \\[ {\\int_{-\\infty}^{\\infty} (L^{x+h}_{t}- L^{x}_{t})^{2} dx- 4ht\\over h^{3/2}} \\stackrel{\\mathcal{L}}{\\to}({64 \\over 3})^{1/2}\\sqrt{\\alpha_{t}} \\eta, \\] as $h\\rar 0$. Equivalently \\[ {\\int_{-\\infty}^{\\infty} (L^{x+1}_{t}- L^{x}_{t})^{2} dx- 4t\\over t^{3/4}} \\stackrel{\\mathcal{L}}{\\to}({64 \\over 3} )^{1/2}\\sqrt{\\alpha_{1}} \\eta, \\] as $t\\rar\\infty$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-01-08T17:55:28.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "brownian local time",
      "continuity",
      "brownian motion"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}