{
  "id": "1801.02999",
  "version": "v1",
  "published": "2018-01-09T15:32:52.000Z",
  "updated": "2018-01-09T15:32:52.000Z",
  "title": "Exact asymptotics for a multi-timescale model",
  "authors": [
    "Mariska Heemskerk",
    "Michel Mandjes"
  ],
  "comment": "17 pages, no figures",
  "categories": [
    "math.PR"
  ],
  "abstract": "In this paper we study the probability $\\xi_n(u):={\\mathbb P}\\left(C_n\\geqslant u n \\right)$, with $C_n:=A(\\psi_n B(\\varphi_n))$ for L\\'{e}vy processes $A(\\cdot)$ and $B(\\cdot)$, and $\\varphi_n$ and $\\psi_n$ non-negative sequences such that $\\varphi_n \\psi_n =n$ and $\\varphi_n\\to\\infty$ as $n\\to\\infty$. Two timescale regimes are distinguished: a `fast' regime in which $\\varphi_n$ is superlinear and a `slow' regime in which $\\varphi_n$ is sublinear. We provide the exact asymptotics of $\\xi_n(u)$ (as $n\\to\\infty$) for both regimes, relying on change-of-measure arguments in combination with Edgeworth-type estimates. The asymptotics have an unconventional form: the exponent contains the commonly observed linear term, but may also contain sublinear terms (the number of which depends on the precise form of $\\varphi_n$ and $\\psi_n$). To showcase the power of our results we include two examples, covering both the case where $C_n$ is lattice and non-lattice.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-01-09T15:32:52.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60F10",
      "60G51",
      "60K37"
    ],
    "keywords": [
      "exact asymptotics",
      "multi-timescale model",
      "contain sublinear terms",
      "edgeworth-type estimates",
      "timescale regimes"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}