{
  "id": "1606.01502",
  "version": "v1",
  "published": "2016-06-05T12:33:55.000Z",
  "updated": "2016-06-05T12:33:55.000Z",
  "title": "An Erdös--Révész type law of the iterated logarithm for order statistics of a stationary Gaussian process",
  "authors": [
    "K. Dębicki",
    "K. M. Kosiński"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $\\{X(t):t\\in\\mathbb R_+\\}$ be a stationary Gaussian process with almost surely (a.s.) continuous sample paths, $\\mathbb E X(t) = 0$, $\\mathbb E X^2(t) = 1$ and correlation function satisfying (i) $r(t) = 1 - C|t|^{\\alpha} + o(|t|^{\\alpha})$ as $t\\to 0$ for some $0\\le\\alpha\\le 2, C>0$, (ii) $\\sup_{t\\ge s}|r(t)|<1$ for each $s>0$ and (iii) $r(t) = O(t^{-\\lambda})$ as $t\\to\\infty$ for some $\\lambda>0$. For any $n\\ge 1$, consider $n$ mutually independent copies of $X$ and denote by $\\{X_{r:n}(t):t\\ge 0\\}$ the $r$th smallest order statistics process, $1\\le r\\le n$. We provide a tractable criterion for assessing whether, for any positive, non-decreasing function $f$, $\\mathbb P(\\mathscr E_f)=\\mathbb P(X_{r:n}(t) > f(t)\\, \\text{i.o.})$ equals 0 or 1. Using this criterion we find that, for a family of functions $f_p(t)$, such that $z_p(t)=\\mathbb P(\\sup_{s\\in[0,1]}X_{r:n}(s)>f_p(t))=O((t\\log^{1-p} t)^{-1})$, $\\mathbb P(\\mathscr E_{f_p})= 1_{\\{p\\ge 0\\}}$. Consequently, with $\\xi_p (t) = \\sup\\{s:0\\le s\\le t, X_{r:n}(s)\\ge f_p(s)\\}$, for $p\\ge 0$, $\\lim_{t\\to\\infty}\\xi_p(t)=\\infty$ and $\\limsup_{t\\to\\infty}(\\xi_p(t)-t)=0$ a.s.. Complementary, we prove an Erd\\\"os-R\\'ev\\'esz type law of the iterated logarithm lower bound on $\\xi_p(t)$, i.e., $\\liminf_{t\\to\\infty}(\\xi_p(t)-t)/h_p(t) = -1$ a.s., $p>1$, $\\liminf_{t\\to\\infty}\\log(\\xi_p(t)/t)/(h_p(t)/t) = -1$ a.s., $p\\in(0,1]$, where $h_p(t)=(1/z_p(t))p\\log\\log t$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-06-05T12:33:55.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60F15",
      "60G70",
      "60G22"
    ],
    "keywords": [
      "stationary gaussian process",
      "erdös-révész type law",
      "iterated logarithm",
      "th smallest order statistics process"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}