{
  "id": "1902.09633",
  "version": "v1",
  "published": "2019-02-25T21:54:38.000Z",
  "updated": "2019-02-25T21:54:38.000Z",
  "title": "Bifractional Brownian motion for $H>1$ and $2HK\\le 1$",
  "authors": [
    "Anna Talarczyk"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "Bifractional Brownian motion on $\\mathbb{R}_+$ is a two parameter centered Gaussian process with covariance function: $$ R_{H,K} (t,s)=\\frac 1{2^K}\\left(\\left(t^{2H}+s^{2H}\\right)^K-\\vert t-s\\vert^{2HK}\\right), \\qquad s,t\\ge 0. $$ This process has been originally introduced by Houdr\\'e and Villa (2002) for the range of parameters $H\\in (0,1]$ and $K\\in (0,1]$. Since then, the range of parameters, for which $R_{H,K}$ is known to be nonnegative definite has been somewhat extended, but the full range is still not known. In the present paper we give an elementary proof that $R_{H,K}$ is nonnegative definite for parameters $H,K$ satisfying $H>1$ and $0<2HK\\le 1$. We show that $R_{H,K}$ can be decomposed into a sum of two nonnegative definite functions. As a side product we also obtain a decomposition of the fractional Brownian motion with Hurst parameter $H<\\frac 12$ into a sum of time rescaled Brownian motion and another independent self-similar Gaussian process. The main idea relies on decomposition of $R_{H,K}$ with help of the following two functions $$ C_{\\gamma}(s,t)=(s+t)^\\gamma-\\left(\\max (s,t)\\right)^\\gamma, \\qquad s,t\\ge 0 $$ and $$ Q_{\\gamma}(s,t)=\\left(\\max (s,t)\\right)^\\gamma-\\vert t-s\\vert^\\gamma, \\qquad s,t\\ge 0 $$ which are nonnegative definite for $0<\\gamma\\le 1$. We also discuss some simple properties of bifractional Brownian motion with $H>1$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-02-25T21:54:38.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G15",
      "60G18",
      "60G22",
      "42A82"
    ],
    "keywords": [
      "bifractional brownian motion",
      "nonnegative definite",
      "independent self-similar gaussian process",
      "parameter centered gaussian process",
      "time rescaled brownian motion"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}