{
  "id": "0809.1864",
  "version": "v2",
  "published": "2008-09-10T19:58:10.000Z",
  "updated": "2008-11-10T20:14:39.000Z",
  "title": "On the invariant measure of the random difference equation $X_n=A_n X_{n-1}+ B_n$ in the critical case",
  "authors": [
    "Sara Brofferio",
    "Dariusz Buraczewski",
    "Ewa Damek"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "We consider the autoregressive model on $\\R^d$ defined by the following stochastic recursion $X_n = A_n X_{n-1}+B_n$, where $\\{(B_n,A_n)\\}$ are i.i.d. random variables valued in $\\R^d\\times \\R^+$. The critical case, when $\\E\\big[\\log A_1\\big]=0$, was studied by Babillot, Bougeorol and Elie, who proved that there exists a unique invariant Radon measure $\\nu$ for the Markov chain $\\{X_n \\}$. In the present paper we prove that the weak limit of properly dilated measure $\\nu$ exists and defines a homogeneous measure on $\\R^d\\setminus \\{0\\}$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2008-11-10T20:14:39.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J10",
      "60B15",
      "60G50"
    ],
    "keywords": [
      "random difference equation",
      "critical case",
      "invariant measure",
      "unique invariant radon measure",
      "random variables"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0809.1864B"
    }
  }
}