{
  "id": "2210.00420",
  "version": "v1",
  "published": "2022-10-02T04:21:11.000Z",
  "updated": "2022-10-02T04:21:11.000Z",
  "title": "An Improved Berry-Esseen Bound of Least Squares Estimation for Fractional Ornstein-Uhlenbeck Processes",
  "authors": [
    "Yong Chen",
    "Xiangmeng Gu"
  ],
  "comment": "This english version is translated by Hanxiao GENG from a Chinese version submitted to Acta Mathematica Scientia. The readers in China can find the original Chinese version from the authors by email",
  "categories": [
    "math.PR"
  ],
  "abstract": "The aim of this paper is twofold. First, it offers a novel formula to calculate the inner product of the bounded variation function in the Hilbert space $\\mathcal{H}$ associated with the fractional Brownian motion with Hurst parameter $H\\in (0,\\frac12)$. This formula is based on a kind of decomposition of the Lebesgue-Stieljes measure of the bounded variation function and the integration by parts formula of the Lebesgue-Stieljes measure. Second, as an application of the formula, we explore that as $T\\to\\infty$, the asymptotic line for the square of the norm of the bivariate function $f_T(t,s)=e^{-\\theta|t-s|}1_{\\{0\\leq s,t\\leq T\\}}$ in the symmetric tensor space $\\mathcal{H}^{\\odot 2}$ (as a function of $T$), and improve the Berry-Ess\\'{e}en type upper bound for the least squares estimation of the drift coefficient of the fractional Ornstein-Uhlenbeck processes with Hurst parameter $H\\in (\\frac14,\\frac12)$. The asymptotic analysis of the present paper is much more subtle than that of Lemma 17 in Hu, Nualart, Zhou(2019) and the improved Berry-Ess\\'{e}en type upper bound is the best improvement of the result of Theorem 1.1 in Chen, Li (2021). As a by-product, a second application of the above asymptotic analysis is given, i.e., we also show the Berry-Ess\\'{e}en type upper bound for the moment estimation of the drift coefficient of the fractional Ornstein-Uhlenbeck processes where the method is obvious different to that of Proposition 4.1 in Sottinen, Viitasaari(2018).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-10-02T04:21:11.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G15",
      "60G22",
      "62M09"
    ],
    "keywords": [
      "fractional ornstein-uhlenbeck processes",
      "squares estimation",
      "type upper bound",
      "berry-esseen bound",
      "bounded variation function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}