{
  "id": "1412.1068",
  "version": "v1",
  "published": "2014-12-02T20:57:01.000Z",
  "updated": "2014-12-02T20:57:01.000Z",
  "title": "Self-similar scaling limits of Markov chains on the positive integers",
  "authors": [
    "Jean Bertoin",
    "Igor Kortchemski"
  ],
  "comment": "42 pages, 1 figure",
  "categories": [
    "math.PR"
  ],
  "abstract": "We are interested in the asymptotic behavior of Markov chains on the set of positive integers for which, loosely speaking, large jumps are rare and occur at a rate that behaves like a negative power of the current state, and such that small positive and negative steps of the chain roughly compensate each other. If $X_{n}$ is such a Markov chain started at $n$, we establish a limit theorem for $\\frac{1}{n}X_{n}$ appropriately scaled in time, where the scaling limit is given by a nonnegative self-similar Markov process. We also study the asymptotic behavior of the time needed by $X_{n}$ to reach some fixed finite set. We identify three different regimes (roughly speaking the transient, the recurrent and the positive-recurrent regimes) in which $X_{n}$ exhibits different behavior. The present results extend those of Haas & Miermont who focused on the case of non-increasing Markov chains. We further present a number of applications to the study of Markov chains with asymptotically zero drifts such as Bessel-type random walks, nonnegative self-similar Markov processes, invariance principles for random walks conditioned to stay positive, and exchangeable coalescence-fragmentation processes.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-12-02T20:57:01.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "markov chain",
      "self-similar scaling limits",
      "positive integers",
      "asymptotic behavior",
      "bessel-type random walks"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 42,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1412.1068B"
    }
  }
}