{
  "id": "1206.1251",
  "version": "v1",
  "published": "2012-06-06T14:55:10.000Z",
  "updated": "2012-06-06T14:55:10.000Z",
  "title": "Approximation of a random process with variable smoothness",
  "authors": [
    "Enkelejd Hashorva",
    "Mikhail Lifshits",
    "Oleg Seleznjev"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "We consider the rate of piecewise constant approximation to a locally stationary process $X(t),t\\in [0,1]$, having a variable smoothness index $\\alpha(t)$. Assuming that $\\alpha(\\cdot)$ attains its unique minimum at zero and satisfies the regularity condition, we propose a method for construction of observation points (composite dilated design) and find an asymptotics for the integrated mean square error, where a piecewise constant approximation $X_n$ is based on $N(n)\\sim n$ observations of $X$. Further, we prove that the suggested approximation rate is optimal, and then show how to find an optimal constant.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-06-06T14:55:10.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "random process",
      "piecewise constant approximation",
      "integrated mean square error",
      "locally stationary process",
      "approximation rate"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1206.1251H"
    }
  }
}