{
  "id": "math/0408327",
  "version": "v3",
  "published": "2004-08-24T09:33:15.000Z",
  "updated": "2005-11-21T09:54:35.000Z",
  "title": "Annealed deviations of random walk in random scenery",
  "authors": [
    "Nina Gantert",
    "Wolfgang König",
    "Zhan Shi"
  ],
  "comment": "32 pages, revised",
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $(Z_n)_{n\\in\\N}$ be a $d$-dimensional {\\it random walk in random scenery}, i.e., $Z_n=\\sum_{k=0}^{n-1}Y(S_k)$ with $(S_k)_{k\\in\\N_0}$ a random walk in $\\Z^d$ and $(Y(z))_{z\\in\\Z^d}$ an i.i.d. scenery, independent of the walk. The walker's steps have mean zero and finite variance. We identify the speed and the rate of the logarithmic decay of $\\P(\\frac 1n Z_n>b_n)$ for various choices of sequences $(b_n)_n$ in $[1,\\infty)$. Depending on $(b_n)_n$ and the upper tails of the scenery, we identify different regimes for the speed of decay and different variational formulas for the rate functions. In contrast to recent work \\cite{AC02} by A. Asselah and F. Castell, we consider sceneries {\\it unbounded} to infinity. It turns out that there are interesting connections to large deviation properties of self-intersections of the walk, which have been studied recently by X. Chen \\cite{C03}.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2005-11-21T09:54:35.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K37",
      "60F10",
      "60J55"
    ],
    "keywords": [
      "random walk",
      "random scenery",
      "annealed deviations",
      "large deviation properties",
      "logarithmic decay"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 32,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004math......8327G"
    }
  }
}