{
  "id": "1011.6355",
  "version": "v1",
  "published": "2010-11-29T20:21:47.000Z",
  "updated": "2010-11-29T20:21:47.000Z",
  "title": "Exact asymptotics of supremum of a stationary Gaussian process over a random interval",
  "authors": [
    "Marek Arendarczyk",
    "Krzysztof Debicki"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $\\{X(t) : t \\in [0, \\infty) \\}$ be a centered stationary Gaussian process. We study the exact asymptotics of $\\pr (\\sup_{s \\in [0,T]} X(t) > u)$, as $u \\to \\infty$, where $T$ is an independent of \\{X(t)\\} nonnegative random variable. It appears that the heaviness of $T$ impacts the form of the asymptotics, leading to three scenarios: the case of integrable $T$, the case of $T$ having regularly varying tail distribution with parameter $\\lambda\\in(0,1)$ and the case of $T$ having slowly varying tail distribution.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-11-29T20:21:47.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G15",
      "60G70",
      "68M20"
    ],
    "keywords": [
      "exact asymptotics",
      "random interval",
      "centered stationary gaussian process",
      "slowly varying tail distribution",
      "regularly varying tail distribution"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1011.6355A"
    }
  }
}