{
  "id": "1806.05702",
  "version": "v1",
  "published": "2018-06-14T18:48:59.000Z",
  "updated": "2018-06-14T18:48:59.000Z",
  "title": "On (signed) Takagi-Landsberg functions: $p^{\\text{th}}$ variation, maximum, and modulus of continuity",
  "authors": [
    "Yuliya Mishura",
    "Alexander Schied"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "We study a class $\\mathfrak X^H$ of signed Takagi-Landsberg functions with Hurst parameter $H\\in(0,1)$. We first show that the functions in $\\mathfrak X^H$ admit a linear $p^{\\text{th}}$ variation along the sequence of dyadic partitions of $[0,1]$, where $p=1/H$. The slope of the linear increase can be represented as the $p^{\\text{th}}$ absolute moment of the infinite Bernoulli convolution with parameter $2^{H-1}$. The existence of a continuous $p^{\\text{th}}$ variation enables the use of the functions in $\\mathfrak X^H$ as test integrators for higher-order pathwise It\\^o calculus. Our next results concern the maximum, the maximizers, and the modulus of continuity of the classical Takagi-Landsberg function for all $0<H<1$. Then we identify the uniform maximum, the uniform maximal oscillation, and a uniform modulus of continuity for the class $\\mathfrak X^H$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-06-14T18:48:59.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28A80",
      "26A30",
      "26A27",
      "60G17",
      "60H05",
      "26A15",
      "26A16"
    ],
    "keywords": [
      "continuity",
      "uniform maximal oscillation",
      "infinite bernoulli convolution",
      "hurst parameter",
      "uniform maximum"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}