{
  "id": "1807.03575",
  "version": "v1",
  "published": "2018-07-10T11:38:34.000Z",
  "updated": "2018-07-10T11:38:34.000Z",
  "title": "Strong renewal theorems and local large deviations for multivariate random walks and renewals",
  "authors": [
    "Quentin Berger"
  ],
  "comment": "45 pages, comments are welcome",
  "categories": [
    "math.PR"
  ],
  "abstract": "We study a random walk $\\mathbf{S}_n$ on $\\mathbb{Z}^d$ ($d\\geq 1$), in the domain of attraction of an operator-stable distribution with index $\\boldsymbol{\\alpha}=(\\alpha_1,\\ldots,\\alpha_d) \\in (0,2]^d$: in particular, we allow the scalings to be different along the different coordinates. We prove a strong renewal theorem, $i.e.$ a sharp asymptotic of the Green function $G(\\mathbf{0},\\mathbf{x})$ as $\\|\\mathbf{x}\\|\\to +\\infty$, along the \"favorite direction or scaling\": (i) if $\\sum_{i=1}^d \\alpha_i^{-1} < 2$ (reminiscent of Garcia-Lamperti's condition when $d=1$ [Comm. Math. Helv. $\\mathbf{37}$, 1963]); (ii) if a certain $local$ condition holds (reminiscent of Doney's condition [Probab. Theory Relat. Fields $\\mathbf{107}$, 1997] when $d=1$). We also provide uniform bounds on the Green function $G(\\mathbf{0},\\mathbf{x})$, sharpening estimates when $\\mathbf{x}$ is away from this favorite direction or scaling. These results improve significantly the existing literature, which was mostly concerned with the case $\\alpha_i\\equiv \\alpha$, in the favorite scaling, and has even left aside the case $\\alpha\\in[1,2)$ with non-zero mean. Most of our estimates rely on new general (multivariate) local large deviations results, that were missing in the literature and that are of interest on their own.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-07-10T11:38:34.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K05",
      "60G50",
      "60F15",
      "60F10"
    ],
    "keywords": [
      "strong renewal theorem",
      "multivariate random walks",
      "green function",
      "local large deviations results",
      "favorite direction"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 45,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}