{
  "id": "1310.6661",
  "version": "v2",
  "published": "2013-10-24T16:35:21.000Z",
  "updated": "2014-08-12T18:09:22.000Z",
  "title": "A note on the times of first passage for `nearly right-continuous' random walks",
  "authors": [
    "Matija Vidmar"
  ],
  "comment": "10 pages, 1 figure",
  "categories": [
    "math.PR"
  ],
  "abstract": "A natural extension of a right-continuous integer-valued random walk is one which can jump to the right by one or two units. First passage times above a given fixed level then admit a tractable Laplace transform (probability generating function). Explicit expressions for the probabilities that the respective overshoots are either $0$ or $1$, according as the random walk crosses a given level for the first time either continuously or not, also obtain. An interesting non-obvious observation, which follows from the analysis, is that any such (non-degenerate) random walk will, eventually in $n\\in \\mathbb{N}\\cup \\{0\\}$, always be more likely to pass over the level $n$ for the first time with overshoot zero, rather than one. Some applications are considered.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-08-12T18:09:22.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G50",
      "60G51"
    ],
    "keywords": [
      "first time",
      "first passage times",
      "random walk crosses",
      "tractable laplace transform",
      "probability generating function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1310.6661V"
    }
  }
}