{
  "id": "1911.00695",
  "version": "v1",
  "published": "2019-11-02T11:09:35.000Z",
  "updated": "2019-11-02T11:09:35.000Z",
  "title": "Berry-Esseen bounds for random projections of $\\ell_p^n$-balls",
  "authors": [
    "Samuel G. G. Johnston",
    "Joscha Prochno"
  ],
  "comment": "23 pages",
  "categories": [
    "math.PR",
    "math.FA"
  ],
  "abstract": "In this work we study the rate of convergence in the central limit theorem for the Euclidean norm of random orthogonal projections of vectors chosen at random from an $\\ell_p^n$-ball which has been obtained in [Alonso-Guti\\'errez, Prochno, Th\\\"ale: Gaussian fluctuations for high-dimensional random projections of $\\ell_p^n$-balls, Bernoulli 25(4A), 2019, 3139--3174]. More precisely, for any $n\\in\\mathbb N$ let $E_n$ be a random subspace of dimension $k_n\\in\\{1,\\ldots,n\\}$, $P_{E_n}$ the orthogonal projection onto $E_n$, and $X_n$ be a random point in the unit ball of $\\ell_p^n$. We prove a Berry-Esseen theorem for $\\|P_{E_n}X_n\\|_2$ under the condition that $k_n\\to\\infty$. This answers in the affirmative a conjecture of Alonso-Guti\\'errez, Prochno, and Th\\\"ale who obtained a rate of convergence under the additional condition that $k_n/n^{2/3}\\to\\infty$ as $n\\to\\infty$. In addition, we study the Gaussian fluctuations and Berry-Esseen bounds in a $3$-fold randomized setting where the dimension of the Grassmannian is also chosen randomly. Comparing deterministic and randomized subspace dimensions leads to a quite interesting observation regarding the central limit behavior. In this work we also discuss the rate of convergence in the central limit theorem of [Kabluchko, Prochno, Th\\\"ale: High-dimensional limit theorems for random vectors in $\\ell_p^n$-balls, Commun. Contemp. Math. (2019)] for general $\\ell_q$-norms of non-projected vectors chosen at random in an $\\ell_p^n$-ball.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-11-02T11:09:35.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60F05",
      "41A25",
      "52A20",
      "52A23",
      "52A22"
    ],
    "keywords": [
      "berry-esseen bounds",
      "central limit theorem",
      "vectors chosen",
      "gaussian fluctuations",
      "central limit behavior"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}