{
  "id": "1603.07214",
  "version": "v1",
  "published": "2016-03-23T15:00:23.000Z",
  "updated": "2016-03-23T15:00:23.000Z",
  "title": "The rate of convergence for the renewal theorem in $\\mathbb{R}^d$",
  "authors": [
    "Jean-Baptiste Boyer"
  ],
  "categories": [
    "math.PR",
    "math.DS"
  ],
  "abstract": "Let $\\rho$ be a borelian probability measure on $\\mathrm{SL}\\_d(\\mathbb{R})$. Consider the random walk $(X\\_n)$ on $\\mathbb{R}^d\\setminus\\{0\\}$ defined by $\\rho$ : for any $x\\in \\mathbb{R}^d\\setminus\\{0\\}$, we set $X\\_0 =x$ and $X\\_{n+1} = g\\_{n+1} X\\_n$ where $(g\\_n)$ is an iid sequence of $\\mathrm{SL}\\_d(\\mathbb{R})-$valued random variables of law $\\rho$. Guivarc'h and Raugi proved that under an assumption on the subgroup generated by the support of $\\rho$ (strong irreducibility and proximality), this walk is transient.In particular, this proves that if $f$ is a compactly supported continuous function on $\\mathbb{R}^d$, then the function $Gf(x) :=\\mathbb{E}\\_x \\sum\\_{k=0}^{+\\infty} f(X\\_n)$ is well defined for any $x\\in \\mathbb{R}^d \\setminus\\{0\\}$.Guivarc'h and Le Page proved the renewal theorem in this situation : they study the possible limits of $Gf$ at $0$ and in this article, we study the rate of convergence in their renewal theorem.To do so, we consider the family of operators $(P(it))\\_{t\\in \\mathbb{R}}$ defined for any continuous function $f$ on the sphere $\\mathbb{S}^{d-1}$ and any $x\\in \\mathbb{S}^{d-1}$ by\\[P(it) f(x) = \\int\\_{\\mathrm{SL}\\_d(\\mathbb{R})} e^{-it \\ln \\frac{\\|gx\\|}{\\|x\\|}} f\\left(\\frac{gx}{\\|gx\\|}\\right) \\mathrm{d}\\rho(g)\\]We prove that, adding an exponential moment condition to the strong irreducibility and proximality condition, we have that for some $L\\in \\mathbb{R}$ and any $t\\_0 \\in \\mathbb{R}\\_+^\\ast$,\\[\\sup\\_{\\substack{t\\in \\mathbb{R}\\\\ |t| \\geqslant t\\_0}} \\frac{1}{|t|^L} \\left\\| (I\\_d-P(it))^{-1} \\right\\| \\text{is finite}\\]where the norm is taken in some space of h\\\"older-continuous functions on the sphere.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-03-23T15:00:23.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "renewal theorem",
      "convergence",
      "strong irreducibility",
      "exponential moment condition",
      "borelian probability measure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160307214B"
    }
  }
}