{
  "id": "2004.13440",
  "version": "v1",
  "published": "2020-04-28T11:47:38.000Z",
  "updated": "2020-04-28T11:47:38.000Z",
  "title": "Asymptotics of product of nonnegative 2-by-2 matrices and maximum of a (2,1) random walk with asymptotically zero drift",
  "authors": [
    "Hongyan SUN",
    "Hua-Ming WANG"
  ],
  "comment": "18 pages",
  "categories": [
    "math.PR",
    "math.RA"
  ],
  "abstract": "Consider matrix product $A_kA_{k-1}\\cdots A_1$ with $A_n,n\\ge1$ some nonnegative 2-by-2 matrices. Under certain condition, we show that $\\forall i,j= 1,2,$ $(A_kA_{k-1}\\cdots A_1)_{i,j}\\sim c\\varrho(A_k)\\varrho(A_{k-1})\\cdots \\varrho(A_1)$ as $k\\rightarrow\\infty$ with $\\varrho(A_n)$ the spectral radius of $A_n$ and $0<c<\\infty$ some constant. As an application, consider (2,1) random walk with asymptotically zero drift on the positive half line. Let $M$ be the maximum of an excursion starting from $2$ and ending at the point(either $0$ or $1$), where the walk hits the set $\\{0,1\\}$ for the first time. We study the distribution of $M$ and characterize its asymptotics, which are quite different from those of simple random walks.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-04-28T11:47:38.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "15B48",
      "60G50",
      "60J10"
    ],
    "keywords": [
      "asymptotically zero drift",
      "asymptotics",
      "nonnegative",
      "simple random walks",
      "spectral radius"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}