{
  "id": "0806.4561",
  "version": "v2",
  "published": "2008-06-27T16:38:41.000Z",
  "updated": "2010-09-07T12:38:32.000Z",
  "title": "Moments of exit times from wedges for non-homogeneous random walks with asymptotically zero drifts",
  "authors": [
    "Iain M. MacPhee",
    "Mikhail V. Menshikov",
    "Andrew R. Wade"
  ],
  "comment": "35 pages, 2 figures; most of the material in v1 (which had a different title) appears in revised form in v2 or the companion paper arXiv:0910.1772",
  "journal": "Journal of Theoretical Probability, Vol. 26 (2013), no. 1, p. 1-30",
  "doi": "10.1007/s10959-012-0411-x",
  "categories": [
    "math.PR"
  ],
  "abstract": "We study quantitative asymptotics of planar random walks that are spatially non-homogeneous but whose mean drifts have some regularity. Specifically, we study the first exit time $\\tau_\\alpha$ from a wedge with apex at the origin and interior half-angle $\\alpha$ by a non-homogeneous random walk on the square lattice with mean drift at $x$ of magnitude $O(1/|x|)$ as $|x| \\to \\infty$. This is the critical regime for the asymptotic behaviour: under mild conditions, a previous result of the authors (see arXiv:0910.1772) stated that $\\tau_\\alpha < \\infty$ a.s. for any $\\alpha$ (while for a stronger drift field $\\tau_\\alpha$ is infinite with positive probability). Here we study the more difficult problem of the existence and non-existence of moments $E[\\tau_\\alpha^s]$, $s>0$. Assuming (in common with much of the literature) a uniform bound on the walk's increments, we show that for $\\alpha < \\pi/2$ there exists $s_0 \\in (0,\\infty)$ such that $E[\\tau_\\alpha^s]$ is finite for $s < s_0$ but infinite for $s > s_0$; under specific assumptions on the drift field we show that we can attain $E[\\tau_\\alpha^s] = \\infty$ for any $s > 1/2$. We show that for $\\alpha \\leq \\pi$ there is a phase transition between drifts of magnitude $O(1/|x|)$ (the critical regime) and $o(1/|x|)$ (the subcritical regime). In the subcritical regime we obtain a non-homogeneous random walk analogue of a theorem for Brownian motion due to Spitzer, under considerably weaker conditions than those previously given (including work by Varopoulos) that assumed zero drift.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2010-09-07T12:38:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J10",
      "60G40",
      "60G50"
    ],
    "keywords": [
      "non-homogeneous random walk",
      "asymptotically zero drifts",
      "mean drift",
      "random walk analogue",
      "critical regime"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 35,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0806.4561M"
    }
  }
}