{
  "id": "1007.0155",
  "version": "v3",
  "published": "2010-07-01T12:55:13.000Z",
  "updated": "2011-02-09T14:46:59.000Z",
  "title": "Convergence of the all-time supremum of a Lévy process in the heavy-traffic regime",
  "authors": [
    "Kamil Marcin Kosinski",
    "Onno Boxma",
    "Bert Zwart"
  ],
  "journal": "Queueing Systems 67 (2011) 295-304",
  "doi": "10.1007/s11134-011-9215-4",
  "categories": [
    "math.PR"
  ],
  "abstract": "In this paper we derive a technique of obtaining limit theorems for suprema of L\\'evy processes from their random walk counterparts. For each $a>0$, let $\\{Y^{(a)}_n:n\\ge 1\\}$ be a sequence of independent and identically distributed random variables and $\\{X^{(a)}_t:t\\ge 0\\}$ be a L\\'evy processes such that $X_1^{(a)}\\stackrel{d}{=} Y_1^{(a)}$, $\\mathbb E X_1^{(a)}<0$ and $\\mathbb E X_1^{(a)}\\uparrow0$ as $a\\downarrow0$. Let $S^{(a)}_n=\\sum_{k=1}^n Y^{(a)}_k$. Then, under some mild assumptions, $\\Delta(a)\\max_{n\\ge 0} S_n^{(a)}\\stackrel{d}{\\to} R\\iff\\Delta(a)\\sup_{t\\ge 0} X^{(a)}_t\\stackrel{d}{\\to} R$, for some random variable $R$ and some function $\\Delta(\\cdot)$. We utilize this result to present a number of limit theorems for suprema of L\\'evy processes in the heavy-traffic regime.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2011-02-09T14:46:59.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G50",
      "60G51",
      "60K25",
      "60F17"
    ],
    "keywords": [
      "heavy-traffic regime",
      "lévy process",
      "all-time supremum",
      "levy processes",
      "convergence"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1007.0155M"
    }
  }
}