{
  "id": "2105.07978",
  "version": "v3",
  "published": "2021-05-17T16:02:51.000Z",
  "updated": "2022-02-22T10:10:30.000Z",
  "title": "Asymptotic results for certain first-passage times and areas of renewal processes",
  "authors": [
    "Claudio Macci",
    "Barbara Pacchiarotti"
  ],
  "comment": "22 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "We consider the process $\\{x-N(t):t\\geq 0\\}$, where $x\\in\\mathbb{R}_+$ and $\\{N(t):t\\geq 0\\}$ is a renewal process with light-tailed distributed holding times. We are interested in the joint distribution of $(\\tau(x),A(x))$ where $\\tau(x)$ is the first-passage time of $\\{x-N(t):t\\geq 0\\}$ to reach zero or a negative value, and $A(x):=\\int_0^{\\tau(x)}(x-N(t))dt$ is the corresponding first-passage (positive) area swept out by the process $\\{x-N(t):t\\geq 0\\}$. We remark that we can define the sequence $\\{(\\tau(n),A(n)):n\\geq 1\\}$ by referring to the concept of integrated random walk. Our aim is to prove asymptotic results as $x\\to\\infty$ in the fashion of large (and moderate) deviations.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2022-02-22T10:10:30.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "first-passage time",
      "renewal process",
      "asymptotic results",
      "reach zero",
      "joint distribution"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}