{
  "id": "1412.3934",
  "version": "v1",
  "published": "2014-12-12T10:01:06.000Z",
  "updated": "2014-12-12T10:01:06.000Z",
  "title": "Extremes of order statistics of self-similar processes",
  "authors": [
    "Chengxiu Ling"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $\\{X_i(t),t\\ge0\\}, 1\\le i\\le n$ be independent copies of a random process $\\{X(t), t\\ge0\\}$. For a given positive constant $u$, define the set of $r$th conjunctions $C_r(u):=\\{t\\in[0,1]: X_{r:n}(t)>u\\}$ with $ X_{r:n}$ the $r$th largest order statistics of $X_i, 1\\le i\\le n$. In numerical applications such as brain mapping and digital communication systems, of interest is the approximation of $p_r(u)=\\mathbb P\\{C_r(u)\\neq\\phi\\}$. Instead of stationary processes dealt with by D\\c{e}bicki et al. (2014), we consider in this paper $X$ a self-similar $\\mathbb R$-valued process with $P$-continuous sample paths. By imposing the Albin's conditions directly on $X$, we establish an exact asymptotic expansion of $p_r(u)$ as $u$ tends to infinity. As a by-product we derive the asymptotic tail behaviour of the mean sojourn time of $X_{r:n}$ over an increasing threshold. Finally, our findings are illustrated for the case that $X$ is a bi-fractional Brownian motion, a sub-fractional Brownian motion, and a generalized self-similar skew-Gaussian process.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-12-12T10:01:06.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G15",
      "60G70"
    ],
    "keywords": [
      "self-similar processes",
      "th largest order statistics",
      "generalized self-similar skew-gaussian process",
      "sub-fractional brownian motion",
      "mean sojourn time"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1412.3934L"
    }
  }
}