{
  "id": "1711.10202",
  "version": "v1",
  "published": "2017-11-28T09:39:03.000Z",
  "updated": "2017-11-28T09:39:03.000Z",
  "title": "Empirical processes for recurrent and transient random walks in random scenery",
  "authors": [
    "Nadine Guillotin-Plantard",
    "Francoise Pene",
    "Martin Wendler"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "In this paper, we are interested in the asymptotic behaviour of the sequence of processes $(W_n(s,t))_{s,t\\in[0,1]}$ with \\begin{equation*} W_n(s,t):=\\sum_{k=1}^{\\lfloor nt\\rfloor}\\big(1_{\\{\\xi_{S_k}\\leq s\\}}-s\\big) \\end{equation*} where $(\\xi_x, x\\in\\mathbb{Z}^d)$ is a sequence of independent random variables uniformly distributed on $[0,1]$ and $(S_n)_{n\\in\\mathbb N}$ is a random walk evolving in $\\mathbb{Z}^d$, independent of the $\\xi$'s. In Wendler (2016), the case where $(S_n)_{n\\in\\mathbb N}$ is a recurrent random walk in $\\mathbb{Z}$ such that $(n^{-\\frac 1\\alpha}S_n)_{n\\geq 1}$ converges in distribution to a stable distribution of index $\\alpha$, with $\\alpha\\in(1,2]$, has been investigated. Here, we consider the cases where $(S_n)_{n\\in\\mathbb N}$ is either: a) a transient random walk in $\\mathbb{Z}^d$, b) a recurrent random walk in $\\mathbb{Z}^d$ such that $(n^{-\\frac 1d}S_n)_{n\\geq 1}$ converges in distribution to a stable distribution of index $d\\in\\{1,2\\}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-11-28T09:39:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G50",
      "60F17",
      "62G30"
    ],
    "keywords": [
      "transient random walk",
      "random scenery",
      "empirical processes",
      "recurrent random walk",
      "independent random variables"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}