{
  "id": "1411.0423",
  "version": "v1",
  "published": "2014-11-03T10:52:28.000Z",
  "updated": "2014-11-03T10:52:28.000Z",
  "title": "Conditional limit theorems for products of random matrices",
  "authors": [
    "Ion Grama",
    "Emile Le Page",
    "Marc Peigné"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "Consider the product $G_{n}=g_{n} ... g_{1}$ of the random matrices $g_{1},...,g_{n}$ in $GL(d,\\mathbb{R}) $ and the random process $ G_{n}v=g_{n}... g_{1}v$ in $\\mathbb{R}^{d}$ starting at point $v\\in \\mathbb{R}^{d}\\smallsetminus \\{0\\} .$ It is well known that under appropriate assumptions, the sequence $(\\log \\Vert G_{n}v\\Vert)_{n\\geq 1}$ behaves like a sum of i.i.d.\\ r.v.'s and satisfies standard classical properties such as the law of large numbers, law of iterated logarithm and the central limit theorem. Denote by $\\mathbb{B}$ the closed unit ball in $\\mathbb{R}^{d}$ and by $\\mathbb{B}^{c}$ its complement. For any $v\\in \\mathbb{B}^{c}$ define the exit time of the random process $G_{n}v$ from $\\mathbb{B}^{c}$ by $\\tau_{v}=\\min \\{n\\geq 1:G_{n}v\\in \\mathbb{B}\\} .$ We establish the asymptotic as $narrow \\infty $ of the probability of the event $\\{\\tau_{v}>n\\} $ and find the limit law for the quantity $\\frac{1}{\\sqrt{n}} \\log \\Vert G_{n}v\\Vert $ conditioned that $\\tau_{v}>n.$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-11-03T10:52:28.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60B20",
      "60J05",
      "60J45",
      "37A50"
    ],
    "keywords": [
      "conditional limit theorems",
      "random matrices",
      "random process",
      "central limit theorem",
      "satisfies standard classical properties"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1411.0423G"
    }
  }
}