{
  "id": "0804.2378",
  "version": "v1",
  "published": "2008-04-15T13:12:27.000Z",
  "updated": "2008-04-15T13:12:27.000Z",
  "title": "Almost-sure Growth Rate of Generalized Random Fibonacci sequences",
  "authors": [
    "Elise Janvresse",
    "Benoît Rittaud",
    "Thierry De La Rue"
  ],
  "journal": "Annales de l'IHP - Probabilit\\'es et Statistiques 46, 1 (2010) 135-158",
  "doi": "10.1214/09-AIHP312",
  "categories": [
    "math.PR"
  ],
  "abstract": "We study the generalized random Fibonacci sequences defined by their first nonnegative terms and for $n\\ge 1$, $F_{n+2} = \\lambda F_{n+1} \\pm F_{n}$ (linear case) and $\\widetilde F_{n+2} = |\\lambda \\widetilde F_{n+1} \\pm \\widetilde F_{n}|$ (non-linear case), where each $\\pm$ sign is independent and either $+$ with probability $p$ or $-$ with probability $1-p$ ($0<p\\le 1$). Our main result is that, when $\\lambda$ is of the form $\\lambda_k = 2\\cos (\\pi/k)$ for some integer $k\\ge 3$, the exponential growth of $F_n$ for $0<p\\le 1$, and of $\\widetilde F_{n}$ for $1/k < p\\le 1$, is almost surely positive and given by $$ \\int_0^\\infty \\log x d\\nu_{k, \\rho} (x), $$ where $\\rho$ is an explicit function of $p$ depending on the case we consider, taking values in $[0, 1]$, and $\\nu_{k, \\rho}$ is an explicit probability distribution on $\\RR_+$ defined inductively on generalized Stern-Brocot intervals. We also provide an integral formula for $0<p\\le 1$ in the easier case $\\lambda\\ge 2$. Finally, we study the variations of the exponent as a function of $p$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2008-04-15T13:12:27.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37H15",
      "60J05",
      "11J70"
    ],
    "keywords": [
      "generalized random fibonacci sequences",
      "almost-sure growth rate",
      "explicit probability distribution",
      "easier case"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0804.2378J"
    }
  }
}