{
  "id": "1402.2558",
  "version": "v1",
  "published": "2014-02-11T16:45:13.000Z",
  "updated": "2014-02-11T16:45:13.000Z",
  "title": "Non-homogeneous random walks on a semi-infinite strip",
  "authors": [
    "Nicholas Georgiou",
    "Andrew R. Wade"
  ],
  "comment": "27 pages",
  "journal": "Stochastic Processes and their Applications, Vol. 124 (2014), no. 10, p. 3179-3205",
  "doi": "10.1016/j.spa.2014.05.005",
  "categories": [
    "math.PR"
  ],
  "abstract": "We study the asymptotic behaviour of Markov chains $(X_n,\\eta_n)$ on $\\mathbb{Z}_+ \\times S$, where $\\mathbb{Z}_+$ is the non-negative integers and $S$ is a finite set. Neither coordinate is assumed to be Markov. We assume a moments bound on the jumps of $X_n$, and that, roughly speaking, $\\eta_n$ is close to being Markov when $X_n$ is large. This departure from much of the literature, which assumes that $\\eta_n$ is itself a Markov chain, enables us to probe precisely the recurrence phase transitions by assuming asymptotically zero drift for $X_n$ given $\\eta_n$. We give a recurrence classification in terms of increment moment parameters for $X_n$ and the stationary distribution for the large-$X$ limit of $\\eta_n$. In the null case we also provide a weak convergence result, which demonstrates a form of asymptotic independence between $X_n$ (rescaled) and $\\eta_n$. Our results can be seen as generalizations of Lamperti's results for non-homogeneous random walks on $\\mathbb{Z}_+$ (the case where $S$ is a singleton). Motivation arises from modulated queues or processes with hidden variables where $\\eta_n$ tracks an internal state of the system.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-02-11T16:45:13.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J10",
      "60F05",
      "60F15",
      "60K15",
      "60K25"
    ],
    "keywords": [
      "non-homogeneous random walks",
      "semi-infinite strip",
      "markov chain",
      "recurrence phase transitions",
      "increment moment parameters"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1402.2558G"
    }
  }
}