{
  "id": "1803.05682",
  "version": "v1",
  "published": "2018-03-15T10:41:20.000Z",
  "updated": "2018-03-15T10:41:20.000Z",
  "title": "Harmonic functions of random walks in a semigroup via ladder heights",
  "authors": [
    "Irina Ignatiouk-Robert"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "We investigate harmonic functions and the convergence of the sequence of ratios $(P_x(\\tau_\\vartheta {>} n)/P_e(\\tau_\\vartheta {>} n))$ for a random walk on a countable group killed up on the time $\\tau_\\vartheta$ of the first exit from some semi-group with an identity element $e$. Several results of classical renewal theory for one dimensional random walk killed at the first exit from the positive half-line are extended to a multi-dimensional setting. For this purpose, an analogue of the ladder height process and the corresponding renewal function $V$ are introduced. The results are applied to multidimensional random walks killed upon the times of first exit from a convex cone. Our approach combines large deviation estimates and an extension of Choquet-Deny theory.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-03-15T10:41:20.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J10",
      "60J45"
    ],
    "keywords": [
      "harmonic functions",
      "first exit",
      "multidimensional random walks",
      "large deviation estimates",
      "ladder height process"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}