{
  "id": "0708.0649",
  "version": "v2",
  "published": "2007-08-04T22:03:53.000Z",
  "updated": "2016-06-11T02:51:52.000Z",
  "title": "Quenched Limits for Transient, Ballistic, Sub-Gaussian One-Dimensional Random Walk in Random Environment",
  "authors": [
    "Jonathon Peterson"
  ],
  "comment": "28 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "We consider a nearest-neighbor, one-dimensional random walk $\\{X_n\\}_{n\\geq 0}$ in a random i.i.d. environment, in the regime where the walk is transient with speed v_P > 0 and there exists an $s\\in(1,2)$ such that the annealed law of $n^{-1/s} (X_n - n v_P)$ converges to a stable law of parameter s. Under the quenched law (i.e., conditioned on the environment), we show that no limit laws are possible. In particular we show that there exist sequences {t_k} and {t_k'} depending on the environment only, such that a quenched central limit theorem holds along the subsequence t_k, but the quenched limiting distribution along the subsequence t_k' is a centered reverse exponential distribution. This complements the results of a recent paper of Peterson and Zeitouni (arXiv:0704.1778v1 [math.PR]) which handled the case when the parameter $s\\in(0,1)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2007-08-04T22:03:53.000Z",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2016-06-11T02:51:52.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K37",
      "60F05",
      "82C41",
      "82D30"
    ],
    "keywords": [
      "sub-gaussian one-dimensional random walk",
      "random environment",
      "quenched limits",
      "quenched central limit theorem holds"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0708.0649P"
    }
  }
}