{
  "id": "2106.01851",
  "version": "v1",
  "published": "2021-06-03T13:55:54.000Z",
  "updated": "2021-06-03T13:55:54.000Z",
  "title": "Berry-Esséen bounds and almost sure CLT for the quadratic variation of a general Gaussian process",
  "authors": [
    "Yong Chen",
    "Zhen Ding",
    "Ying Li"
  ],
  "comment": "20 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "In this paper, we consider the explicit bound for the second-order approximation of the quadratic variation of a general fractional Gaussian process $(G_t)_{t\\ge 0}$. The second order mixed partial derivative of the covariance function $ R(t,\\, s)=\\mathbb{E}[G_t G_s]$ can be decomposed into two parts, one of which coincides with that of fractional Brownian motion and the other of which is bounded by $(ts)^{H-1}$ up to a constant factor. This condition is valid for a class of continuous Gaussian processes that fails to be self-similar or have stationary increments. %Some examples include the subfractional Brownian motion and the bi-fractional Brownian motion. Under this assumption, we obtain the optimal Berry-Ess\\'{e}en bounds when $H\\in (0,\\,\\frac23]$ and the upper Berry-Ess\\'{e}en bounds when $H\\in (\\frac23,\\,\\frac34]$. As a by-product, we also show the almost sure central limit theorem (ASCLT) for the quadratic variation when $H\\in (0,\\,\\frac34]$. The results extend that of \\cite{NP 09} to the case of general Gaussian processes, unify and improve the Berry-Ess\\'{e}en bounds in \\cite{Tu 11}, \\cite{AE 12} and \\cite{KL 21} for respectively the sub-fractional Brownian motion, the bi-fractional Brownian motion and the sub-bifractional Brownian motion.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-06-03T13:55:54.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60H07",
      "60G15",
      "60F05"
    ],
    "keywords": [
      "general gaussian process",
      "quadratic variation",
      "sure clt",
      "berry-esséen bounds",
      "bi-fractional brownian motion"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}