Berry-Esseen bound for the Moment Estimation of the fractional Ornstein-Uhlenbeck model under fixed step size discrete observations
Zheng Tang, Ying Li, Haili Yang, Hua Yi, Yong Chen
TL;DR
This work derives Berry–Esseen type convergence rates for a fixed-step discrete estimator of the drift parameter in a fractional Ornstein–Uhlenbeck model, showing that $d_{Kol}(\sqrt{n}(\hat{\theta}_n-\theta), \mathcal{N})$ decays at rates $\mathcal{O}(n^{-1/2})$ for $H\in(0,5/8]$ and $\mathcal{O}(n^{-(3-4H)})$ for $H\in(5/8,3/4)$ (with a $\sqrt{n/\log n}$ scaling at $H=3/4$), all under fixed (non-vanishing) observation spacing $h$. The authors develop a framework based on second Wiener chaos analysis, the Fourth Moment Theorem, and Malliavin calculus to obtain these bounds directly for discrete observations, removing previous constraints that required $h_n\to0$. They further extend the results to OU models driven by a broad class of Gaussian noises (sub-fractional, bi-fractional, sub-bifractional, generalized fBm), establishing robustness of the Berry–Esseen bounds across Gaussian drivers. The approach yields sharp finite-sample rates and broad applicability in estimation for Gaussian-driven diffusions.
Abstract
Let the Ornstein-Uhlenbeck process $\{X_t,\,t\geq 0\}$ driven by a fractional Brownian motion $B^H$ described by $d X_t=-θX_t dt+ d B_t^H,\, X_0=0$ with known parameter $H\in (0,\frac34)$ be observed at discrete time instants $t_k=kh, k=1,2,\dots, n $. If $θ>0$ and if the step size $h>0$ is arbitrarily fixed, we derive Berry-Esséen bound for the ergodic type estimator (or say the moment estimator) $\hatθ_n$, i.e., the Kolmogorov distance between the distribution of $\sqrt{n}(\hatθ_n-θ)$ and its limit distribution is bounded by a constant $C_{θ, H,h}$ times $n^{-\frac12}$ and $ n^{4H-3}$ when $H\in (0,\,\frac58]$ and $H\in (\frac58,\,\frac34)$, respectively. This result greatly improve the previous result in literature where $h$ is forced to go zero. Moreover, we extend the Berry-Esseen bound to the Ornstein-Uhlenbeck model driven by a lot of Gaussian noises such as the sub-bifractional Brownian motion and others. A few ideas of the present paper come from Haress and Hu (2021), Sottinen and Viitasaari (2018), and Chen and Zhou (2021).