{
  "id": "2101.01609",
  "version": "v1",
  "published": "2021-01-05T15:51:10.000Z",
  "updated": "2021-01-05T15:51:10.000Z",
  "title": "Local central limit theorem and potential kernel estimates for a class of symmetric heavy-tailted random variables",
  "authors": [
    "Leandro Chiarini",
    "Milton Jara",
    "Wioletta M. Ruszel"
  ],
  "comment": "33 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "In this article, we study a class of heavy-tailed random variables on $\\mathbb{Z}$ in the domain of attraction of an $\\alpha$-stable random variable of index $\\alpha \\in (0,2)$ satisfying a certain expansion of their characteristic function. Our results include sharp convergence rates for the local (stable) central limit theorem of order $n^{- (1+ \\frac{1}{\\alpha})}$, a detailed expansion of the characteristic function of a long-range random walk with transition probability proportional to $|x|^{-(1+\\alpha)}$ and $\\alpha \\in (0,2)$ and furthermore detailed asymptotic estimates of the discrete potential kernel (Green's function) up to order $\\mathcal{O} \\left( |x|^{\\frac{\\alpha-2}{3}+\\varepsilon} \\right)$ for any $\\varepsilon>0$ small enough, when $\\alpha \\in [1,2)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-01-05T15:51:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60E07",
      "60E10",
      "60F05",
      "60G50",
      "60G52",
      "60J45"
    ],
    "keywords": [
      "local central limit theorem",
      "symmetric heavy-tailted random variables",
      "potential kernel estimates"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 33,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}