{
  "id": "1903.00247",
  "version": "v1",
  "published": "2019-03-01T11:19:10.000Z",
  "updated": "2019-03-01T11:19:10.000Z",
  "title": "Antiduality and Möbius monotonicity: Generalized Coupon Collector Problem",
  "authors": [
    "Paweł Lorek"
  ],
  "comment": "29 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "For a given absorbing Markov chain $X^*$ on a finite state space, a chain $X$ is a sharp antidual of $X^*$ if the fastest strong stationary time of $X$ is equal, in distribution, to the absorption time of $X^*$. In this paper we show a systematic way of finding such an antidual based on some partial ordering of the state space. We use a theory of strong stationary duality developed recently for M\\\"obius monotone Markov chains. We give several sharp antidual chains for Markov chain corresponding to a generalized coupon collector problem. As a consequence - utilizing known results on a limiting distribution of the absorption time - we indicate a separation cutoff (with its window size) in several chains. We also present a chain which (under some conditions) has a prescribed stationary distribution and its fastest strong stationary time is distributed as a prescribed mixture of sums of geometric random variables.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-03-01T11:19:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J10",
      "60G40",
      "06A06"
    ],
    "keywords": [
      "generalized coupon collector problem",
      "fastest strong stationary time",
      "möbius monotonicity",
      "markov chain",
      "sharp antidual"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 29,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}