{
  "id": "0806.3126",
  "version": "v2",
  "published": "2008-06-19T16:48:01.000Z",
  "updated": "2008-09-28T06:46:15.000Z",
  "title": "Laws of the iterated logarithm for a class of iterated processes",
  "authors": [
    "Erkan Nane"
  ],
  "comment": "13 pages",
  "journal": "Statistics & Probability Letters, Volume 79, Issue 16, 15 August 2009, Pages 1744-1751",
  "doi": "10.1016/j.spl.2009.04.013",
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $X=\\{X(t), t\\geq 0\\}$ be a Brownian motion or a spectrally negative stable process of index $1<\\a<2$. Let $E=\\{E(t),t\\geq 0\\}$ be the hitting time of a stable subordinator of index $0<\\beta<1$ independent of $X$. We use a connection between $X(E(t))$ and the stable subordinator of index $\\beta/\\a$ to derive information on the path behavior of $X(E_t)$. This is an extension of the connection of iterated Brownian motion and (1/4)-stable subordinator due to Bertoin \\cite{bertoin}. Using this connection, we obtain various laws of the iterated logarithm for $X(E(t))$. In particular, we establish law of the iterated logarithm for local time Brownian motion, $X(L(t))$, where $X$ is a Brownian motion (the case $\\a=2$) and $L(t)$ is the local time at zero of a stable process $Y$ of index $1<\\gamma\\leq 2$ independent of $X$. In this case $E(\\rho t)=L(t)$ with $\\beta=1-1/\\gamma$ for some constant $\\rho>0$. This establishes the lower bound in the law of the iterated logarithm which we could not prove with the techniques of our paper \\cite{MNX}. We also obtain exact small ball probability for $X(E_t)$ using ideas from \\cite{aurzada}.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2008-09-28T06:46:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J65",
      "60K99"
    ],
    "keywords": [
      "iterated logarithm",
      "iterated processes",
      "exact small ball probability",
      "local time brownian motion",
      "stable subordinator"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0806.3126N"
    }
  }
}