{
  "id": "math/0610850",
  "version": "v1",
  "published": "2006-10-27T15:11:49.000Z",
  "updated": "2006-10-27T15:11:49.000Z",
  "title": "Ordered random walks",
  "authors": [
    "Peter Eichelsbacher",
    "Wolfgang Konig"
  ],
  "comment": "26 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "We construct the conditional version of $k$ independent and identically distributed random walks on $\\R$ given that they stay in strict order at all times. This is a generalisation of so-called non-colliding or non-intersecting random walks, the discrete variant of Dyson's Brownian motions, which have been considered yet only for nearest-neighbor walks on the lattice. Our only assumptions are moment conditions on the steps and the validity of the local central limit theorem. The conditional process is constructed as a Doob $h$-transform with some positive regular function $V$ that is strongly related with the Vandermonde determinant and reduces to that function for simple random walk. Furthermore, we prove an invariance principle, i.e., a functional limit theorem towards Dyson's Brownian motions, the continuous analogue.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2006-10-27T15:11:49.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G50",
      "60F17"
    ],
    "keywords": [
      "ordered random walks",
      "dysons brownian motions",
      "local central limit theorem",
      "simple random walk",
      "functional limit theorem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math.....10850E"
    }
  }
}