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

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

作　　者：陈勇[1] 古象盟 Chen Yong;Gu Xiangmeng(School of Mathematics and Statistics,Jiangxi Normal University,Nanchang 330022)

机构地区：[1]江西师范大学数学与统计学院,南昌330022

出　　处：《数学物理学报（A辑）》2023年第3期855-882,共28页Acta Mathematica Scientia

基　　金：国家自然科学基金(11961033)~。

摘　　要：该文目的有二.一是得到了当Hurst参数H∈(0,1/2)时,分数布朗运动联系的Hilbert空间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H^(⊙2)中二元函数fT(t,s)=e^(−θ|t−s|)1{0≤s,t≤T},其范数的平方做为T的函数当T→∞时的渐近线,改进了当H∈(1/4,1/2)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.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 H associated with the fractional Brownian motion with Hurst parameter H∈(0,1/2).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→∞,the asymptotic line for the square of the norm of the bivariate function fT(t,s)=e^(−θ|t−s|)1{0≤s,t≤T}in the symmetric tensor space H^(⊙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∈(1/4,1/2).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).

关 键 词：分数布朗运动 分数 ORNSTEIN-UHLENBECK 过程 Berry-Esséen类上界 

分 类 号：O211.64%[理学—概率论与数理统计]