{
  "id": "1211.1318",
  "version": "v2",
  "published": "2012-11-06T17:26:52.000Z",
  "updated": "2014-02-20T00:38:27.000Z",
  "title": "Logarithmic asymptotics for multidimensional extremes under non-linear scalings",
  "authors": [
    "Kamil Marcin Kosinski",
    "Michel Mandjes"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $\\boldsymbol W=\\{\\boldsymbol W_n:n\\in\\mathbb N\\}$ be a sequence of random vectors in $\\mathbb R^d$, $d\\ge 1$. This paper considers the logarithmic asymptotics of the extremes of $\\boldsymbol W$, that is, for any vector $\\boldsymbol q>\\boldsymbol 0$ in $\\mathbb R^d$, we find $$\\log\\mathbb P\\left(\\exists{n\\in\\mathbb N}:\\boldsymbol W_n> u \\boldsymbol q\\right), \\quad\\text{as} u\\to\\infty.$$ We follow the approach of the restricted large deviation principle introduced in Duffy et al. \\textit{Logarithmic asymptotics for the supremum of a stochastic process} (Ann. Appl. Probab., 13:430--445, 2003). That is, we assume that, for every $\\boldsymbol q\\ge\\boldsymbol 0$, and some scalings $\\{a_n\\},\\{v_n\\}$, $\\frac{1}{v_n}\\log\\mathbb P\\left(\\boldsymbol W_n/a_n\\ge u \\boldsymbol q\\right)$ has a, continuous in $\\boldsymbol q$, limit $J_{\\boldsymbol W}(\\boldsymbol q)$. We allow the scalings $\\{a_n\\}$ and $\\{v_n\\}$ to be regularly varying with a positive index. This approach is general enough to incorporate sequences $\\boldsymbol W$, such that the probability law of $\\boldsymbol W_n/a_n$ satisfies the large deviation principle with continuous, not necessarily convex, rate functions. The formula for these asymptotics agrees with the seminal papers on this topic.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-02-20T00:38:27.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60F10",
      "60G70"
    ],
    "keywords": [
      "logarithmic asymptotics",
      "non-linear scalings",
      "multidimensional extremes",
      "restricted large deviation principle",
      "asymptotics agrees"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1211.1318M"
    }
  }
}