{
  "id": "2103.16430",
  "version": "v1",
  "published": "2021-03-30T15:25:18.000Z",
  "updated": "2021-03-30T15:25:18.000Z",
  "title": "Projections of the uniform distribution on the cube -- a large deviation perspective",
  "authors": [
    "Samuel G. G. Johnston",
    "Zakhar Kabluchko",
    "Joscha Prochno"
  ],
  "comment": "13 pages",
  "categories": [
    "math.PR",
    "math.FA"
  ],
  "abstract": "For $n\\in\\mathbb N$ let $\\Theta^{(n)}$ be a random vector uniformly distributed on the unit sphere $\\mathbb S^{n-1}$, and consider the associated random probability measure $\\mu_{\\Theta^{(n)}}$ given by setting \\[ \\mu_{\\Theta^{(n)}}(A) := \\mathbb{P} \\left[ \\langle U, \\Theta^{(n)} \\rangle \\in A \\right],\\qquad U \\sim \\text{Unif}([-1,1]^n) \\] for Borel subets $A$ of $\\mathbb{R}$. It is known that the sequence of random probability measures $\\mu_{\\Theta^{(n)}}$ converges weakly to the centered Gaussian distribution with variance $1/3$. We prove a large deviation principle for the sequence $\\mu_{\\Theta^{(n)}}$ with speed $n$ and explicit good rate rate function given by $I(\\nu(\\alpha)) := - \\frac{1}{2} \\log ( 1 - ||\\alpha||_2^2)$ whenever $\\nu(\\alpha)$ is the law of a random variable of the form \\begin{align*} \\sqrt{1 - ||\\alpha||_2^2 } \\frac{Z}{\\sqrt 3} + \\sum_{ k = 1}^\\infty \\alpha_k U_k, \\end{align*} where $Z$ is standard Gaussian independent of $U_1,U_2,\\ldots$ which are i.i.d.\\ $\\text{Unif}[-1,1]$, and $\\alpha_1 \\geq \\alpha_2 \\geq \\ldots $ is a non-increasing sequence of non-negative reals with $||\\alpha||_2<1$. We obtain a similar result for projections of the uniform distribution on the discrete cube $\\{-1,+1\\}^n$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-03-30T15:25:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60F10",
      "46B06",
      "52A23"
    ],
    "keywords": [
      "uniform distribution",
      "large deviation perspective",
      "projections",
      "associated random probability measure",
      "rate rate function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}