{
  "id": "1806.11498",
  "version": "v1",
  "published": "2018-06-29T15:58:12.000Z",
  "updated": "2018-06-29T15:58:12.000Z",
  "title": "On the upper bound of the $L_p$ discrepancy of Halton's sequence and the Central Limit Theorem for Hammersley's net",
  "authors": [
    "Mordechay B. Levin"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $(H_s(n))_{n \\geq 1}$ be an $s-$dimensional Halton's sequence, and let ${\\mathcal{H}}_{s+1,N}=(H_s(n),n/N)_{n=0}^{N-1}$ be the $s+1-$dimensional Hammersley point set. Let $D(\\mathbf{x},(H_n)_{n=0}^{N-1} )$ be the local discrepancy of $(H_n)_{n=0}^{N-1}$, and let $D_{s,p} ( (H_n)_{n=0}^{N-1}) $ be the $L_p$ discrepancy of $(H_n)_{n=0}^{N-1} $. It is known that $\\limsup_{N \\to \\infty} N (\\log N)^{-s/2} D_{s,p} (H_s(N))_{n=0}^{N-1} >0$. In this paper, we prove that $$D_{s,p} ((H_s(N))_{n=0}^{N-1}) = O(N^{-1} \\log^{s/2} N) \\quad {\\rm for} \\; \\; N \\to \\infty.$$ I.e., we found the smallest possible order of magnitude of $L_p$ discrepancy of Halton's sequence. Then we prove the Central Limit Theorem for Hammersley net : \\begin{equation}\\nonumber N^{-1} D(\\bar{\\mathbf{x}},\\mathcal{H}_{s+1,N} )/ D_{s+1,2}(\\mathcal{H}_{s+1,N}) \\stackrel{w}{\\rightarrow} \\mathcal{N}(0,1), \\end{equation} where $\\bar{\\mathbf{x}}$ is a uniformly distributed random variable in $[0,1]^{s+1}$. The main tool is the theorem on $p$-adic logarithmic forms.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-06-29T15:58:12.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11K38"
    ],
    "keywords": [
      "central limit theorem",
      "upper bound",
      "hammersleys net",
      "discrepancy",
      "dimensional hammersley point set"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}