{
  "id": "1001.4872",
  "version": "v1",
  "published": "2010-01-27T08:45:43.000Z",
  "updated": "2010-01-27T08:45:43.000Z",
  "title": "The asymptotic behavior of densities related to the supremum of a stable process",
  "authors": [
    "R. A. Doney",
    "M. S. Savov"
  ],
  "comment": "Published in at http://dx.doi.org/10.1214/09-AOP479 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)",
  "journal": "Annals of Probability 2010, Vol. 38, No. 1, 316-326",
  "doi": "10.1214/09-AOP479",
  "categories": [
    "math.PR"
  ],
  "abstract": "If $X$ is a stable process of index $\\alpha\\in(0,2)$ whose L\\'{e}vy measure has density $cx^{-\\alpha-1}$ on $(0,\\infty)$, and $S_1=\\sup_{0<t\\leq1}X_t$, it is known that $P(S_1>x)\\backsim A\\alpha ^{-1}x^{-\\alpha}$ as $x\\to\\infty$ and $P(S_1\\leq x)\\backsim B\\alpha^{-1}\\rho^{-1}x^{\\alpha\\rho}$ as $x\\downarrow0$. [Here $\\rho =P(X_1>0)$ and $A$ and $B$ are known constants.] It is also known that $S_1$ has a continuous density, $m$ say. The main point of this note is to show that $m(x)\\backsim Ax^{-(\\alpha+1)}$ as $x\\to\\infty$ and $m(x)\\backsim Bx^{\\alpha\\rho-1}$ as $x\\downarrow0$. Similar results are obtained for related densities.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-01-27T08:45:43.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J30",
      "60F15"
    ],
    "keywords": [
      "stable process",
      "asymptotic behavior",
      "similar results",
      "main point"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}