{
  "id": "1509.01704",
  "version": "v1",
  "published": "2015-09-05T14:48:19.000Z",
  "updated": "2015-09-05T14:48:19.000Z",
  "title": "Renewal approximation for the absorption time of a decreasing Markov chain",
  "authors": [
    "Gerold Alsmeyer",
    "Alexander Marynych"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "We consider a Markov chain $(M_{n})_{n\\ge 0}$ on the set $\\mathbb{N}_{0}$ of nonnegative integers which is eventually decreasing, i.e. $\\mathbb{P}\\{M_{n+1}<M_{n}|M_{n}\\ge a\\}=1$ for some $a\\in\\mathbb{N}$ and all $n\\ge 0$. We are interested in the asymptotic behaviour of the law of the stopping time $T=T(a):=\\inf\\{k\\in\\mathbb{N}_{0}: M_{k}<a\\}$ under $\\mathbb{P}_{n}:=\\mathbb{P}(\\cdot|M_{0}=n)$ as $n\\to\\infty$. Assuming that the decrements of $(M_{n})_{n\\ge 0}$ given $M_{0}=n$ possess a kind of stationarity for large $n$, we derive sufficient conditions for the convergence in minimal $L^{p}$-distance of $\\mathbb{P}_{n}((T-a_{n})/b_{n}\\in\\cdot)$ to some non-degenerate, proper law and give an explicit form of the constants $a_{n}$ and $b_{n}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-09-05T14:48:19.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60F05",
      "60J10"
    ],
    "keywords": [
      "decreasing markov chain",
      "absorption time",
      "renewal approximation",
      "asymptotic behaviour",
      "derive sufficient conditions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150901704A"
    }
  }
}