{
  "id": "1810.10145",
  "version": "v1",
  "published": "2018-10-24T01:02:02.000Z",
  "updated": "2018-10-24T01:02:02.000Z",
  "title": "Sojourn times of Gaussian processes with trend",
  "authors": [
    "Krzysztof Debicki",
    "Peng Liu",
    "Zbigniew Michna"
  ],
  "comment": "25 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "We derive exact tail asymptotics of sojourn time above the level $u\\geq 0$ $$ \\mathbb{P}\\left(v(u)\\int_0^T \\mathbb{I}(X(t)-ct>u)d t>x\\right), \\quad x\\geq 0 $$ as $u\\to\\infty$, where $X$ is a Gaussian process with continuous sample paths, $c>0$, $v(u)$ is a positive function of $u$ and $T\\in (0,\\infty]$. Additionally, we analyze asymptotic distributional properties of $$\\tau_u(x):=\\inf\\left\\{t\\geq 0: v(u)\\frac{1}{\\Delta_I(u)} \\int_0^t \\mathbb{I}(X(s)-cs>u)d s>x\\right\\}, $$ as $u\\to\\infty$, $x\\geq 0$, where $\\inf\\emptyset=\\infty$. The findings of this contribution are illustrated by a detailed analysis of a class of Gaussian processes with stationary increments.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-10-24T01:02:02.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G15",
      "60G70"
    ],
    "keywords": [
      "gaussian process",
      "sojourn time",
      "analyze asymptotic distributional properties",
      "derive exact tail asymptotics",
      "stationary increments"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}