{
  "id": "0909.1084",
  "version": "v1",
  "published": "2009-09-06T14:19:23.000Z",
  "updated": "2009-09-06T14:19:23.000Z",
  "title": "A CLT for the $L^{2}$ norm of increments of local times of Lévy processes as time goes to infinity",
  "authors": [
    "Michael B. Marcus",
    "Jay Rosen"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $X=\\{X_{t},t\\in R_{+}\\}$ be a symmetric L\\'{e}vy process with local time $\\{L^{x}_{t} ; (x,t)\\in R^{1}\\times R^{1}_{+}\\}$. When the L\\'{e}vy exponent $\\psi(\\la)$ is regularly varying at zero with index $1<\\beta\\leq 2$, and satisfies some additional regularity conditions, \\begin{eqnarray*} && {\\int_{-\\infty}^{\\infty} (L^{x+1}_{t}- L^{x}_{t})^{2} dx- E(\\int_{-\\infty}^{\\infty} (L^{x+1}_{t}- L^{x}_{t})^{2} dx)\\over t\\sqrt{\\psi^{-1}(1/t)}}\\label{r5.0tweaksabs} && \\stackrel{\\mathcal{L}}{\\Longrightarrow}(8c_{\\psi,1 })^{1/2}(\\int_{-\\infty}^{\\finfty} (L_{\\beta,1}^{x})^{2} dx)^{1/2} \\eta \\end{eqnarray*} as $t\\rar\\infty$, where $L_{\\bb,1}=\\{L^{x}_{\\beta, 1} ; x \\in R^{1} \\}$ denotes the local time, at time 1, of a symmetric stable process with index $\\beta$, $\\eta$ is a normal random variable with mean zero and variance one that is independent of $L_{\\beta,1}$, and $c_{\\psi,1}$ is a known constant that depends on $\\psi$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-09-06T14:19:23.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60F05",
      "60J55",
      "60G51"
    ],
    "keywords": [
      "local time",
      "lévy processes",
      "increments",
      "additional regularity conditions",
      "mean zero"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0909.1084M"
    }
  }
}