{
  "id": "2012.14004",
  "version": "v1",
  "published": "2020-12-27T19:39:39.000Z",
  "updated": "2020-12-27T19:39:39.000Z",
  "title": "Central Limit Theorem for $(t,s)$-sequences, I",
  "authors": [
    "Mordechay B. Levin"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $ (X_n)_{n \\geq 0} $ be a digital $(t,s)$-sequence in base $2$, $\\mathcal{P}_m =(X_n)_{n=0}^{2^m-1} $, and let $D(\\mathcal{P}_m, Y )$ be the local discrepancy of $\\mathcal{P}_m$. Let $T \\oplus Y$ be the digital addition of $T$ and $Y$, and let $$\\mathcal{M}_{s,p} (\\mathcal{P}_m) =\\Big( \\int_{[0,1)^{2s}} |D(\\mathcal{P}_m \\oplus T , Y ) |^p \\mathrm{d}T \\mathrm{d}Y \\Big)^{1/p} .$$ In this paper, we prove that $D(\\mathcal{P}_m \\oplus T , Y ) / \\mathcal{M}_{s,2} (\\mathcal{P}_m)$ weakly converge to the standard Gaussisian distribution for $m \\rightarrow \\infty$, where $T,Y$ are uniformly distributed random variables in $[0,1)^s$. In addition, we prove that \\begin{equation} \\nonumber \\mathcal{M}_{s,p} (\\mathcal{P}_m) / \\mathcal{M}_{s,2} (\\mathcal{P}_m) \\to \\frac{1}{\\sqrt{2\\pi}}\\int_{-\\infty}^{\\infty} |u|^p e^{-u^2/2} \\mathrm{d}u \\quad {\\rm for} \\; \\; m \\to \\infty , \\;\\; p>0. \\end{equation}",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-12-27T19:39:39.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11K38"
    ],
    "keywords": [
      "central limit theorem",
      "standard gaussisian distribution",
      "local discrepancy",
      "uniformly distributed random variables",
      "digital addition"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}