{
  "id": "1805.04802",
  "version": "v1",
  "published": "2018-05-13T01:03:44.000Z",
  "updated": "2018-05-13T01:03:44.000Z",
  "title": "Exact asymptotic formulae of the stationary distribution of a discrete-time 2d-QBD process: an example and additional proofs",
  "authors": [
    "Toshihisa Ozawa",
    "Masahiro Kobayashi"
  ],
  "comment": "22 pages, 2 figures",
  "categories": [
    "math.PR"
  ],
  "abstract": "A discrete-time two-dimensional quasi-birth-and-death (2d-QBD) process, $\\{{\\boldsymbol{Y}}_n\\}=\\{(X_{1,n},X_{2,n},J_n)\\}$, is a two-dimensional skip-free random walk $\\{(X_{1,n},X_{2,n})\\}$ on $\\mathbb{Z}_+^2$ with a supplemental process $\\{J_n\\}$ on a finite set $S_0$. The supplemental process $\\{J_n\\}$ is called a phase process. The 2d-QBD process $\\{{\\boldsymbol{Y}}_n\\}$ is a Markov chain in which the transition probabilities of the two-dimensional process $\\{(X_{1,n},X_{2,n})\\}$ are modulated depending on the state of the phase process $\\{J_n\\}$. This modulation is assumed to be space homogeneous except for the boundaries of $\\mathbb{Z}_+^2$. Under certain conditions, the directional exact asymptotic formulae of the stationary distribution of the 2d-QBD process have been obtained in Ozawa and Kobayashi [7]. In this paper, we give an example of 2d-QBD process and additional proofs of some lemmas and propositions stated in Ozawa and Kobayashi [7].",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-05-13T01:03:44.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J10",
      "60K25"
    ],
    "keywords": [
      "discrete-time 2d-qbd process",
      "stationary distribution",
      "additional proofs",
      "two-dimensional skip-free random walk",
      "supplemental process"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}