Berry-Esseen bound for the Moment Estimation of the fractional Ornstein-Uhlenbeck model under fixed step size discrete observations

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).