{
  "id": "1403.7354",
  "version": "v2",
  "published": "2014-03-28T12:10:34.000Z",
  "updated": "2014-08-06T16:25:38.000Z",
  "title": "Extremes of Order Statistics of Stationary Processes",
  "authors": [
    "Krzysztof Debicki",
    "Enkelejd Hashorva",
    "Lanpeng Ji",
    "Chengxiu Ling"
  ],
  "comment": "20 pages, revised version",
  "categories": [
    "math.PR",
    "stat.ME",
    "stat.OT"
  ],
  "abstract": "Let $\\{X_i(t),t\\ge0\\}, 1\\le i\\le n$ be independent copies of a stationary process $\\{X(t), t\\ge0\\}$. For given positive constants $u,T$, define the set of $r$th conjunctions $ C_{r,T,u}:= \\{t\\in [0,T]: X_{r:n}(t) > u\\}$ with $X_{r:n}(t)$ the $r$th largest order statistics of $X_1(t), \\ldots , X_n(t), t\\ge 0$. In numerous applications such as brain mapping and digital communication systems, of interest is the approximation of the probability that the set of conjunctions $C_{r,T,u}$ is not empty. Imposing the Albin's conditions on $X$, in this paper we obtain an exact asymptotic expansion of this probability as $u$ tends to infinity. Further, we establish the tail asymptotics of the supremum of a generalized skew-Gaussian process and a Gumbel limit theorem for the minimum order statistics of stationary Gaussian processes. As a by-product we derive a version of Li and Shao's normal comparison lemma for the minimum and the maximum of Gaussian random vectors.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-08-06T16:25:38.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "stationary processes",
      "shaos normal comparison lemma",
      "th largest order statistics",
      "exact asymptotic expansion",
      "stationary gaussian processes"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1403.7354D"
    }
  }
}