{
  "id": "2003.02954",
  "version": "v1",
  "published": "2020-03-05T22:46:57.000Z",
  "updated": "2020-03-05T22:46:57.000Z",
  "title": "Exact asymptotics of component-wise extrema of two-dimensional Brownian motion",
  "authors": [
    "Krzysztof Debicki",
    "Lanpeng Ji",
    "Tomasz Rolski"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "We derive the exact asymptotics of \\[ P\\left( \\sup_{t\\ge 0} \\Bigl( X_1(t) - \\mu_1 t\\Bigr)> u, \\ \\sup_{s\\ge 0} \\Bigl( X_2(s) - \\mu_2 s\\Bigr)> u \\right), \\ \\ u\\to\\infty, \\] where $(X_1(t),X_2(s))_{t,s\\ge0}$ is a correlated two-dimensional Brownian motion with correlation $\\rho\\in[-1,1]$ and $\\mu_1,\\mu_2>0$. It appears that the play between $\\rho$ and $\\mu_1,\\mu_2$ leads to several types of asymptotics. Although the exponent in the asymptotics as a function of $\\rho$ is continuous, one can observe different types of prefactor functions depending on the range of $\\rho$, which constitute a phase-type transition phenomena.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-03-05T22:46:57.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "exact asymptotics",
      "component-wise extrema",
      "correlated two-dimensional brownian motion",
      "phase-type transition phenomena",
      "prefactor functions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}