{
  "id": "1004.4908",
  "version": "v1",
  "published": "2010-04-27T20:17:18.000Z",
  "updated": "2010-04-27T20:17:18.000Z",
  "title": "On convex hull of Gaussian samples",
  "authors": [
    "Yu. Davydov"
  ],
  "comment": "10 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $X_i = {X_i(t), t \\in T}$ be i.i.d. copies of a centered Gaussian process $X = {X(t), t \\in T}$ with values in $\\mathbb{R}^d$ defined on a separable metric space $T.$ It is supposed that $X$ is bounded. We consider the asymptotic behaviour of convex hulls $$ W_n = \\conv\\ {X_1(t), X_n(t), t \\in T}$$ and show that with probability 1 $$ \\lim_{n\\to \\infty} \\frac{1}{\\sqrt{2\\ln n}} W_n = W $$ (in the sense of Hausdorff distance), where the limit shape $W$ is defined by the covariance structure of $X$: $W = \\conv {}\\{K_t, t\\in T}, K_t$ being the concentration ellipsoid of $X(t).$ The asymptotic behavior of the mathematical expectations $Ef(W_n)$, where $f$ is an homogeneous functional is also studied.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-04-27T20:17:18.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "convex hull",
      "gaussian samples",
      "separable metric space",
      "asymptotic behavior",
      "asymptotic behaviour"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1004.4908D"
    }
  }
}