{
  "id": "math/0611860",
  "version": "v1",
  "published": "2006-11-28T12:38:39.000Z",
  "updated": "2006-11-28T12:38:39.000Z",
  "title": "How do random Fibonacci sequences grow?",
  "authors": [
    "Elise Janvresse",
    "Benoît Rittaud",
    "Thierry De La Rue"
  ],
  "journal": "Probability Theory and Related Fields Volume 142, 3-4 (2008) 619-648",
  "doi": "10.1007/s00440-007-0117-7",
  "categories": [
    "math.PR"
  ],
  "abstract": "We study two kinds of random Fibonacci sequences defined by $F_1=F_2=1$ and for $n\\ge 1$, $F_{n+2} = F_{n+1} \\pm F_{n}$ (linear case) or $F_{n+2} = |F_{n+1} \\pm F_{n}|$ (non-linear case), where each sign is independent and either + with probability $p$ or - with probability $1-p$ ($0<p\\le 1$). Our main result is that the exponential growth of $F_n$ for $0<p\\le 1$ (linear case) or for $1/3\\le p\\le 1$ (non-linear case) is almost surely given by $$\\int_0^\\infty \\log x d\\nu_\\alpha (x), $$ where $\\alpha$ is an explicit function of $p$ depending on the case we consider, and $\\nu_\\alpha$ is an explicit probability distribution on $\\RR_+$ defined inductively on Stern-Brocot intervals. In the non-linear case, the largest Lyapunov exponent is not an analytic function of $p$, since we prove that it is equal to zero for $0<p\\le1/3$. We also give some results about the variations of the largest Lyapunov exponent, and provide a formula for its derivative.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2006-11-28T12:38:39.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37H15",
      "60J05",
      "11A55"
    ],
    "keywords": [
      "random fibonacci sequences grow",
      "largest lyapunov exponent",
      "non-linear case",
      "explicit probability distribution",
      "exponential growth"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math.....11860J"
    }
  }
}