{
  "id": "1403.7372",
  "version": "v2",
  "published": "2014-03-28T13:39:52.000Z",
  "updated": "2014-03-31T07:49:10.000Z",
  "title": "Local limit theorem for the maximum of a random walk",
  "authors": [
    "Johannes Kugler"
  ],
  "comment": "19 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "Consider a family of $\\Delta$-latticed aperiodic random walks $\\{S^{(a)},0\\le a\\le a_0\\}$ with increments $X_i^{(a)}$ and non-positive drift $-a$. Suppose that $\\sup_{a\\le a_0}\\mathbf{E}[(X^{(a)})^2]<\\infty$ and $\\sup_{a\\le a_0}\\mathbf{E}[\\max\\{0,X^{(a)}\\}^{2+\\varepsilon}]<\\infty$ for some $\\varepsilon>0$. Assume that $X^{(a)}\\xrightarrow[]{w} X^{(0)}$ as $a\\to 0$ and denote by $M^{(a)}=\\max_{k\\ge 0} S_k^{(a)}$ the maximum of the random walk $S^{(a)}$. In this paper we provide the asymptotics of $\\mathbf{P}(M^{(a)}=y\\Delta)$ as $a\\to 0$ in the case, when $y\\to \\infty$ and $ay=O(1)$. This asymptotics follows from a representation of $\\mathbf{P}(M^{(a)}=y\\Delta)$ via a geometric sum and a uniform renewal theorem, which is also proved in this paper.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-03-31T07:49:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G50",
      "60G70",
      "60K05"
    ],
    "keywords": [
      "local limit theorem",
      "uniform renewal theorem",
      "latticed aperiodic random walks",
      "geometric sum",
      "asymptotics"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1403.7372K"
    }
  }
}