{
  "id": "1707.01331",
  "version": "v1",
  "published": "2017-07-05T11:39:35.000Z",
  "updated": "2017-07-05T11:39:35.000Z",
  "title": "Asymptotics of the order statistics for a process with a regenerative structure",
  "authors": [
    "Natalia Soja-Kukieła"
  ],
  "comment": "12 pages, 1 figure",
  "categories": [
    "math.PR"
  ],
  "abstract": "In the paper, a regenerative process $\\{X_n:n\\in\\mathbb{N}\\}$ with finite mean cycle length is considered. For~$M_n^{(q)}$ denoting the $q$-th largest value in $\\{X_k : 1\\leqslant k \\leqslant n\\}$, we prove that \\begin{equation*} \\sup_{x\\in\\mathbb{R}} \\left|P\\left(M^{(q)}_n\\leqslant x\\right) - G(x)^n \\sum_{k=0}^{q-1}\\frac{\\left(-\\log G(x)^n\\right)^k}{k!}\\gamma_{q,k}(x)\\right| \\to 0,\\quad \\text{as} \\quad n\\to\\infty, \\end{equation*} for $G$ and $\\gamma_{q,k}$ expressed in terms of maxima over the cycle. The result is illustrated with examples.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-07-05T11:39:35.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G70",
      "60K99",
      "60J05"
    ],
    "keywords": [
      "order statistics",
      "regenerative structure",
      "asymptotics",
      "finite mean cycle length",
      "th largest value"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}