{
  "id": "2102.13513",
  "version": "v1",
  "published": "2021-02-26T14:43:40.000Z",
  "updated": "2021-02-26T14:43:40.000Z",
  "title": "Sharp Asymptotics for $q$-Norms of Random Vectors in High-Dimensional $\\ell_p^n$-Balls",
  "authors": [
    "Tom Kaufmann"
  ],
  "comment": "26 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "Sharp large deviation results of Bahadur & Ranga Rao-type are provided for the $q$-norm of random vectors distributed on the $\\ell_p^n$-ball $\\mathbb{B}^n_p$ according to the cone probability measure or the uniform distribution for $1 \\le q<p < \\infty$, thereby furthering previous large deviation results by Kabluchko, Prochno and Th\\\"ale in the same setting. These results are then applied to deduce sharp asymptotics for intersection volumes of different $\\ell_p^n$-balls in the spirit of Schechtman and Schmuckenschl\\\"ager, and for the length of the projection of an $\\ell_p^n$-ball onto a line with uniform random direction. The sharp large deviation results are proven by providing convenient probabilistic representations of the $q$-norms, employing local limit theorems to approximate their densities, and then using geometric results for asymptotic expansions of Laplace integrals of Adriani & Baldi and Liao & Ramanan to integrate over the densities and derive concrete probability estimates.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-02-26T14:43:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "52A23",
      "60F10",
      "46B09",
      "60D05"
    ],
    "keywords": [
      "random vectors",
      "sharp large deviation results",
      "high-dimensional",
      "deduce sharp asymptotics",
      "derive concrete probability estimates"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}