{
  "id": "0912.4550",
  "version": "v2",
  "published": "2009-12-23T02:21:05.000Z",
  "updated": "2010-09-03T00:57:57.000Z",
  "title": "Excursions and local limit theorems for Bessel-like random walks",
  "authors": [
    "Kenneth S. Alexander"
  ],
  "comment": "44 pages. Numerous small corrections and clarifications. References added",
  "categories": [
    "math.PR"
  ],
  "abstract": "We consider reflecting random walks on the nonnegative integers with drift of order 1/x at height x. We establish explicit asymptotics for various probabilities associated to such walks, including the distribution of the hitting time of 0 and first return time to 0, and the probability of being at a given height k at time n (uniformly in a large range of k.) In particular, for drift of form -\\delta/2x + o(1/x) with \\delta > -1, we show that the probability of a first return to 0 at time n is asymptotically n^{-c}\\phi(n), where c = (3+\\delta)/2 and \\phi is a slowly varying function given explicitly in terms of the o(1/x) terms.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2010-09-03T00:57:57.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J10",
      "60J80"
    ],
    "keywords": [
      "local limit theorems",
      "bessel-like random walks",
      "excursions",
      "first return time",
      "probability"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 44,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0912.4550A"
    }
  }
}