{
  "id": "1610.00881",
  "version": "v1",
  "published": "2016-10-04T07:46:01.000Z",
  "updated": "2016-10-04T07:46:01.000Z",
  "title": "Heavy-tailed random walks on complexes of half-lines",
  "authors": [
    "Mikhail V. Menshikov",
    "Dimitri Petritis",
    "Andrew R. Wade"
  ],
  "comment": "35 pages, 2 figures",
  "categories": [
    "math.PR"
  ],
  "abstract": "We study a random walk on a complex of finitely many half-lines joined at a common origin; jumps are heavy-tailed and of two types, either one-sided (towards the origin) or two-sided (symmetric). Transmission between half-lines via the origin is governed by an irreducible Markov transition matrix, with associated stationary distribution $\\mu_k$. If $\\chi_k$ is $1$ for one-sided half-lines $k$ and $1/2$ for two-sided half-lines, and $\\alpha_k$ is the tail exponent of the jumps on half-line $k$, we show that the recurrence classification for the case where all $\\alpha_k \\chi_k \\in (0,1)$ is determined by the sign of $\\sum_k \\mu_k \\cot ( \\chi_k \\pi \\alpha_k )$. In the case of two half-lines, the model fits naturally on $\\mathbb{R}$ and is a version of the oscillating random walk of Kemperman. In that case, the cotangent criterion for recurrence becomes linear in $\\alpha_1$ and $\\alpha_2$; our general setting exhibits the essential non-linearity in the cotangent criterion. For the general model, we also show existence and non-existence of polynomial moments of return times. Our moments results are sharp (and new) for several cases of the oscillating random walk; they are apparently even new for the case of a homogeneous random walk on $\\mathbb{R}$ with symmetric increments of tail exponent $\\alpha \\in (1,2)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-10-04T07:46:01.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J05",
      "60J10",
      "60G50"
    ],
    "keywords": [
      "heavy-tailed random walks",
      "oscillating random walk",
      "cotangent criterion",
      "tail exponent",
      "irreducible markov transition matrix"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 35,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}