{
  "id": "1112.0658",
  "version": "v1",
  "published": "2011-12-03T14:54:06.000Z",
  "updated": "2011-12-03T14:54:06.000Z",
  "title": "Renewal theorems for random walks in random scenery",
  "authors": [
    "Nadine Guillotin-Plantard",
    "Françoise Pène"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "Random walks in random scenery are processes defined by $Z_n:=\\sum_{k=1}^n\\xi_{X_1+...+X_k}$, where $(X_k,k\\ge 1)$ and $(\\xi_y,y\\in\\mathbb Z)$ are two independent sequences of i.i.d. random variables. We suppose that the distributions of $X_1$ and $\\xi_0$ belong to the normal domain of attraction of strictly stable distributions with index $\\alpha\\in[1,2]$ and $\\beta\\in(0,2)$ respectively. We are interested in the asymptotic behaviour as $|a|$ goes to infinity of quantities of the form $\\sum_{n\\ge 1}{\\mathbb E}[h(Z_n-a)]$ (when $(Z_n)_n$ is transient) or $\\sum_{n\\ge 1}{\\mathbb E}[h(Z_n)-h(Z_n-a)]$ (when $(Z_n)_n$ is recurrent) where $h$ is some complex-valued function defined on $\\mathbb{R}$ or $\\mathbb{Z}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-12-03T14:54:06.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "random walks",
      "random scenery",
      "renewal theorems",
      "random variables",
      "independent sequences"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1112.0658G"
    }
  }
}