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 $d X_t=-\theta X_t dt+ d B_t^H,\, X_0=0$ with known parameter $H\in (0,\frac34)$ be observed at discrete time instants $t_k=kh, k=1,2,\dots, n $. If $\theta>0$ and if the step size $h>0$ is arbitrarily fixed, we derive Berry-Ess\'{e}en 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 $n^{-\frac12}$ 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).