{
  "id": "math/0607406",
  "version": "v1",
  "published": "2006-07-18T05:33:02.000Z",
  "updated": "2006-07-18T05:33:02.000Z",
  "title": "Law of Large Numbers for products of random matrices with coefficients in the max-plus semi-ring",
  "authors": [
    "Glenn Merlet"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "We analyze the asymptotic behavior of random variables $x(n,x\\_0)$ defined by $x(0,x\\_0)=x\\_0$ and $x(n+1,x\\_0)=A(n)x(n,x\\_0)$, where $\\sAn$ is a stationary and ergodic sequence of random matrices with entries in the semi-ring \\mbox{$\\R\\cup\\{-\\infty\\}$} whose addition is the $\\max$ and whose multiplication is $+$. Such sequences modelize a large class of discrete event systems, among which timed event graphs, 1-bounded Petri nets, some queuing networks, train or computer networks. We give necessary conditions for $(\\frac{1}{n}x(n,x\\_0))\\_{n\\in\\N}$ to converge almost surely. Then, we prove a general scheme to give partial converse theorems. When $\\max\\_{A\\_{ij}(0)\\neq -\\infty}|A\\_{ij}(0)|$ is integrable, it allows us: - to give a necessary and sufficient condition for the convergence of $(\\frac{1}{n}x(n,0))\\_{n\\in\\N}$ when the sequence $(A(n))\\_{n\\in\\N}$ is i.i.d., - to prove that, if $(A(n) )\\_{n\\in\\N}$ satisfy a condition of reinforced ergodicity and a condition of fixed structure (i.e. $\\P(A\\_{ij}(0)=-\\infty)\\in\\{0,1\\}$), then $(\\frac{1}{n}x(n,0))\\_{n\\in\\N}$ converges almost-surely, - and to reprove the convergence of $(\\frac{1}{n}x(n,0))\\_{n\\in\\N}$ if the diagonal entries are never $-\\infty$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2006-07-18T05:33:02.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60F15",
      "90B15",
      "93C65",
      "93E15"
    ],
    "keywords": [
      "random matrices",
      "large numbers",
      "max-plus semi-ring",
      "coefficients",
      "partial converse theorems"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math......7406M"
    }
  }
}