{
  "id": "0902.4784",
  "version": "v1",
  "published": "2009-02-27T09:55:39.000Z",
  "updated": "2009-02-27T09:55:39.000Z",
  "title": "Integrated functionals of normal and fractional processes",
  "authors": [
    "Boris Buchmann",
    "Ngai Hang Chan"
  ],
  "comment": "Published in at http://dx.doi.org/10.1214/08-AAP531 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)",
  "journal": "Annals of Applied Probability 2009, Vol. 19, No. 1, 49-70",
  "doi": "10.1214/08-AAP531",
  "categories": [
    "math.PR"
  ],
  "abstract": "Consider $Z^f_t(u)=\\int_0^{tu}f(N_s) ds$, $t>0$, $u\\in[0,1]$, where $N=(N_t)_{t\\in\\mathbb{R}}$ is a normal process and $f$ is a measurable real-valued function satisfying $Ef(N_0)^2<\\infty$ and $Ef(N_0)=0$. If the dependence is sufficiently weak Hariz [J. Multivariate Anal. 80 (2002) 191--216] showed that $Z_t^f/t^{1/2}$ converges in distribution to a multiple of standard Brownian motion as $t\\to\\infty$. If the dependence is sufficiently strong, then $Z_t/(EZ_t(1)^2)^{1/2}$ converges in distribution to a higher order Hermite process as $t\\to\\infty$ by a result by Taqqu [Wahrsch. Verw. Gebiete 50 (1979) 53--83]. When passing from weak to strong dependence, a unique situation encompassed by neither results is encountered. In this paper, we investigate this situation in detail and show that the limiting process is still a Brownian motion, but a nonstandard norming is required. We apply our result to some functionals of fractional Brownian motion which arise in time series. For all Hurst indices $H\\in(0,1)$, we give their limiting distributions. In this context, we show that the known results are only applicable to $H<3/4$ and $H>3/4$, respectively, whereas our result covers $H=3/4$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-02-27T09:55:39.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60F05",
      "60F17",
      "60G15",
      "60J65",
      "62E20",
      "62F12"
    ],
    "keywords": [
      "fractional processes",
      "integrated functionals",
      "higher order hermite process",
      "distribution",
      "standard brownian motion"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0902.4784B"
    }
  }
}