{
  "id": "1905.05693",
  "version": "v1",
  "published": "2019-05-14T16:12:15.000Z",
  "updated": "2019-05-14T16:12:15.000Z",
  "title": "Multidimensional random walks conditioned to stay ordered via generalized ladder height functions",
  "authors": [
    "Osvaldo Angtuncio-Hernández"
  ],
  "comment": "18 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "Random walks conditioned to stay positive are a prominent topic in fluctuation theory. One way to construct them is as a random walk conditioned to stay positive up to time $n$, and let $n$ tend to infinity. A second method is conditioning instead to stay positive up to an independent geometric time, and send its parameter to zero. The multidimensional case (condition the components of a $d$-dimensional random walk to be ordered) was solved in [EK08] using the first approach, but some moment conditions need to be imposed. Our approach is based on the second method, which has the advantage to require a minimal restriction, needed only for the finiteness of the $h$-transform in certain cases. We also characterize when the limit is Markovian or sub-Markovian, and give several reexpresions of the $h$-function. Under some conditions given in [Ign18], it can be proved that our $h$-function is the only harmonic function which is zero outside the Weyl chamber $\\{x=(x_1,\\ldots, x_d)\\in \\mathbb{R}^d: x_1<\\cdots < x_d\\}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-05-14T16:12:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G50",
      "60J10"
    ],
    "keywords": [
      "generalized ladder height functions",
      "multidimensional random walks",
      "stay positive",
      "second method",
      "independent geometric time"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}