{
  "id": "1306.6746",
  "version": "v1",
  "published": "2013-06-28T08:11:37.000Z",
  "updated": "2013-06-28T08:11:37.000Z",
  "title": "Joint asymptotic distribution of certain path functionals of the reflected process",
  "authors": [
    "Aleksandar Mijatovic",
    "Martijn Pistorius"
  ],
  "comment": "21 pages, 1 figure",
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $\\tau(x)$ be the first time the reflected process $Y$ of a Levy processes $X$ crosses x>0. The main aim of the paper is to investigate the asymptotic dependence of the path functionals: $Y(t) = X(t) - \\inf_{0\\leq s\\leq t}X(s)$, $M(t,x)=\\sup_{0\\leq s\\leq t}Y(s)-x$ and $Z(x)=Y(\\tau(x))-x$. We prove that under Cramer's condition on X(1), the functionals $Y(t)$, $M(t,y)$ and $Z(x+y)$ are asymptotically independent as $\\min\\{t,y,x\\}\\to\\infty$. We also characterise the law of the limiting overshoot $Z(\\infty)$ of the reflected process. If, as $\\min\\{t,x\\}\\to\\infty$, the quantity $t\\te{-\\gamma x}$ has a positive limit ($\\gamma$ denotes the Cram\\'er coefficient), our results together with the theorem of Doney & Maller (2005) imply the existence and the explicit form of the joint weak limit $(Y(\\infty),M(\\infty),Z(\\infty))$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-06-28T08:11:37.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G51",
      "60F05",
      "60G17"
    ],
    "keywords": [
      "joint asymptotic distribution",
      "reflected process",
      "path functionals",
      "joint weak limit",
      "explicit form"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1306.6746M"
    }
  }
}