{
  "id": "1802.09832",
  "version": "v1",
  "published": "2018-02-27T11:32:32.000Z",
  "updated": "2018-02-27T11:32:32.000Z",
  "title": "Estimates of Potential functions of random walks on $Z$ with zero mean and infinite variance and their applications",
  "authors": [
    "Kohei Uchiyama"
  ],
  "comment": "31 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "We consider an irreducible random walk on the one dimensional integer lattice with zero mean, infinite variance and i.i.d. increments $X_n$ and obtain certain asymptotic properties of the potential function, $a(x)$, of the walk; in particular we show that as $x\\to\\infty$ $$a(x) \\asymp \\frac{x}{m_-(x)} \\quad\\mbox{and}\\quad \\frac{a(-x)}{a(x)} \\to 0 \\quad\\;\\;\\mbox{if}\\quad \\lim_{x\\to +\\infty} \\frac{m_+(x)}{m_-(x)} =0, $$ where $m_\\pm(x) = \\int_0^xdy\\int_y^\\infty P[\\pm X_1>u]du$. The results are applied in order to obtain a sufficient condition for the relative stability of the ladder height and estimates of some escape probabilities from the origin.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-02-27T11:32:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G50",
      "60J45"
    ],
    "keywords": [
      "zero mean",
      "infinite variance",
      "potential function",
      "applications",
      "dimensional integer lattice"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 31,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}