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

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 $H\in (0,\frac{1}{2})$. 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 $T\to \infty$, the asymptotic line for the square of the norm of the bivariate function $f_T(t,s)=e^{-\theta |t-s|}{1}_{\{0\le s,t\le T\}}$ in the symmetric tensor space ${\mathcal{H}}^{\odot 2}$ (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 $H\in (\frac{1}{4},\frac{1}{2})$. 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).

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