{
  "id": "0910.1772",
  "version": "v1",
  "published": "2009-10-09T14:58:22.000Z",
  "updated": "2009-10-09T14:58:22.000Z",
  "title": "Angular asymptotics for multi-dimensional non-homogeneous random walks with asymptotically zero drift",
  "authors": [
    "Iain M. MacPhee",
    "Mikhail V. Menshikov",
    "Andrew R. Wade"
  ],
  "comment": "35 pages, 2 figures (1 colour)",
  "journal": "Markov Processes and Related Fields, Vol. 16 (2010), no. 2, p. 351-388",
  "categories": [
    "math.PR"
  ],
  "abstract": "We study the first exit time $\\tau$ from an arbitrary cone with apex at the origin by a non-homogeneous random walk (Markov chain) on $\\Z^d$ ($d \\geq 2$) with mean drift that is asymptotically zero. Specifically, if the mean drift at $\\bx \\in \\Z^d$ is of magnitude $O(\\| \\bx\\|^{-1})$, we show that $\\tau<\\infty$ a.s. for any cone. On the other hand, for an appropriate drift field with mean drifts of magnitude $\\| \\bx\\|^{-\\beta}$, $\\beta \\in (0,1)$, we prove that our random walk has a limiting (random) direction and so eventually remains in an arbitrarily narrow cone. The conditions imposed on the random walk are minimal: we assume only a uniform bound on 2nd moments for the increments and a form of weak isotropy. We give several illustrative examples, including a random walk in random environment model.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-10-09T14:58:22.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J10",
      "60F15",
      "60K37"
    ],
    "keywords": [
      "multi-dimensional non-homogeneous random walks",
      "asymptotically zero drift",
      "angular asymptotics",
      "mean drift",
      "appropriate drift field"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 35,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0910.1772M"
    }
  }
}