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 $\{X_t,t\ge 0\}$ driven by a fractional Brownian motion $B^H$ described by $d{X}_t=-\theta X_{t}dt+d{B}_t^H,X_0=0$ with known parameter $H\in (0,\frac{3}{4})$ be observed at discrete time instants $t_k=kh,k=1,2,\dots ,n$. If $\theta >0$ 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) ${\hat{\theta }}_n$, i.e., the Kolmogorov distance between the distribution of $\sqrt{n}({\hat{\theta }}_n-\theta )$ and its limit distribution is bounded by a constant $C_{\theta ,H,h}$ times $n^{-\frac{1}{2}}$ and $n^{4H-3}$ when $H\in (0,\frac{5}{8}]$ and $H\in (\frac{5}{8},\frac{3}{4})$, 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).