{
  "id": "math/0508261",
  "version": "v2",
  "published": "2005-08-15T15:18:25.000Z",
  "updated": "2005-08-16T04:43:30.000Z",
  "title": "Laws of the iterated logarithm for α-time Brownian motion",
  "authors": [
    "Erkan Nane"
  ],
  "comment": "30 pages",
  "journal": "Electronic Journal of Probability, 11 (2006), 434-459.",
  "categories": [
    "math.PR"
  ],
  "abstract": "We introduce a class of iterated processes called $\\alpha$-time Brownian motion for $0<\\alpha \\leq 2$. These are obtained by taking Brownian motion and replacing the time parameter with a symmetric $\\alpha$-stable process. We prove a Chung-type law of the iterated logarithm (LIL) for these processes which is a generalization of LIL proved in \\cite{hu} for iterated Brownian motion. When $\\alpha =1$ it takes the following form $$ \\liminf_{T\\to\\infty}T^{-1/2}(\\log \\log T) \\sup_{0\\leq t\\leq T}|Z_{t}|=\\pi^{2}\\sqrt{\\lambda_{1}} a.s. $$ where $\\lambda_{1}$ is the first eigenvalue for the Cauchy process in the interval $[-1,1].$ We also define the local time $L^{*}(x,t)$ and range $R^{*}(t)=|\\{x: Z(s)=x \\text{for some} s\\leq t\\}|$ for these processes for $1<\\alpha <2$. We prove that there are universal constants $c_{R},c_{L}\\in (0,\\infty) $ such that $$ \\limsup_{t\\to\\infty}\\frac{R^{*}(t)}{(t/\\log \\log t)^{1/2\\alpha}\\log \\log t}= c_{R} a.s. $$ $$ \\liminf_{t\\to\\infty} \\frac{\\sup_{x\\in \\RR{R}}L^{*}(x,t)}{(t/\\log \\log t)^{1-1/2\\alpha}}= c_{L} a.s. $$",
  "revisions": [
    {
      "version": "v2",
      "updated": "2005-08-16T04:43:30.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J65",
      "60K99"
    ],
    "keywords": [
      "iterated logarithm",
      "time brownian motion",
      "time parameter",
      "chung-type law",
      "universal constants"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math......8261N"
    }
  }
}