{
  "id": "1608.02175",
  "version": "v1",
  "published": "2016-08-07T02:40:53.000Z",
  "updated": "2016-08-07T02:40:53.000Z",
  "title": "Precise large deviations of the first passage time",
  "authors": [
    "Dariusz Buraczewski",
    "Mariusz Maślanka"
  ],
  "comment": "10 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $S_n$ be partial sums of an i.i.d. sequence $\\{X_i\\}$. We assume that $\\mathbb{E} X_1 <0$ and $\\mathbb{P}[X_1>0]>0$. In this paper we study the first passage time $$ \\tau_u = \\inf\\{n:\\; S_n > u\\}. $$ The classical Cram\\'er's estimate of the ruin probability says that $$ \\mathbb{P}[\\tau_u<\\infty] \\sim C e^{-\\alpha_0 u}\\quad \\text{as } u\\to \\infty, $$ for some parameter $\\alpha_0$. The aim of the paper is to describe precise large deviations of the first crossing by $S_n$ a linear boundary, more precisely for a fixed parameter $\\rho$ we study asymptotic behavior of $\\mathbb{P}\\left[\\tau_u = \\lfloor u/\\rho\\rfloor \\right]$ as $u$ tends to infinity.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-08-07T02:40:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G50",
      "60F10"
    ],
    "keywords": [
      "first passage time",
      "precise large deviations",
      "ruin probability says",
      "study asymptotic behavior",
      "classical cramers estimate"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}