分数Ornstein-Uhlenbeck 过程最小二乘估计改进的Berry-Esséen 界

数学物理学报 ›› 2023, Vol. 43 ›› Issue (3): 855-882.

分数Ornstein-Uhlenbeck 过程最小二乘估计改进的Berry-Esséen 界

陈勇^*(),古象盟()

江西师范大学数学与统计学院南昌 330022

收稿日期:2022-09-05 修回日期:2023-02-09 出版日期:2023-06-26 发布日期:2023-06-01
通讯作者: 陈勇 E-mail:chenyong77@gmail.com;zhishi@pku.org.cn
作者简介:古象盟, E-mail: zhishi@pku.org.cn
基金资助:
国家自然科学基金(11961033)

An Improved Berry-Esséen Bound of Least Squares Estimation for Fractional Ornstein-Uhlenbeck Processes

Chen Yong^*(),Gu Xiangmeng()

School of Mathematics and Statistics, Jiangxi Normal University, Nanchang 330022

Received:2022-09-05 Revised:2023-02-09 Online:2023-06-26 Published:2023-06-01
Contact: Yong Chen E-mail:chenyong77@gmail.com;zhishi@pku.org.cn
Supported by:
NSFC(11961033)

摘要/Abstract

摘要：

该文目的有二. 一是得到了当Hurst参数 $H\in(0,\frac12)$时, 分数布朗运动联系的Hilbert 空间${\cal H}$中有界变差函数的一种新颖的内积计算公式. 这个新公式基于有界变差函数的Lebesgue-Stieljes测度的一种分解以及Lebesgue-Stieljes测度的分部积分公式. 二是作为该公式的应用, 通过寻找对称张量空间${\cal H}^{\odot 2}$ 中二元函数$f_T(t,s)=e^{-\theta|t-s|}1_{\{0\leq s,t\leq T\}}$, 其范数的平方做为$T$的函数当$T\to \infty$时的渐近线, 改进了当 $H\in(\frac14,\frac12)$时, 分数Ornstein-Uhlenbeck 过程漂移系数最小二乘估计的 Berry-Esséen类上界. 该文的渐近分析比Hu, Nualart, Zhou (2019)引理17 的相应结论精细许多; 该文改进的Berry-Esséen界是Chen, Li (2021)定理1.1 相应结论的最佳改进. 作为一个附产品, 该文也给出上述渐近分析的另一个应用, 分数Ornstein-Uhlenbeck 过程漂移系数矩估计的 Berry-Esséen类上界, 其证明方法和Sottinen, Viitasaari (2018) 命题4.1 的方法显著不同.

关键词: 分数布朗运动, 分数 Ornstein-Uhlenbeck 过程, Berry-Esséen类上界

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 ${\cal H}$ 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 ${\cal H}^{\odot 2}$ (as a function of $T$), and improve the Berry-Esséen 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-Esséen 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-Esséen 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).

Key words: Fractional Brownian motion, Fractional Ornstein-Uhlenbeck process, Berry-Esséen bound

中图分类号:

O211.64%

陈勇,古象盟. 分数Ornstein-Uhlenbeck 过程最小二乘估计改进的Berry-Esséen 界[J]. 数学物理学报, 2023, 43(3): 855-882.

Chen Yong,Gu Xiangmeng. An Improved Berry-Esséen Bound of Least Squares Estimation for Fractional Ornstein-Uhlenbeck Processes[J]. Acta mathematica scientia,Series A, 2023, 43(3): 855-882.

图/表 2

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