Abstract:

This paper studies the statistical inference problem for the unknown drift parameter $\theta$ of an Ornstein-Uhlenbeck process driven by a class of Gaussian processes, defined by $d \zeta_t=-\theta \zeta_t d t+\sigma d G_t, \quad t\in[0,T],$ under discrete observations at points $t_j = j\Delta_n$, $j = 1, \dots, n$, where $\Delta_n$ represents the observation step size, $n$ is the sample size, and $T_n = n\Delta_n$ denotes the entire observation interval. We first construct a moment estimator $\theta$. Then, the strong consistency and asymptotic normality of the moment estimator $\tilde{\theta}_{n}$ are established when the Hurst index $H \in (0, \frac{3}{4})$. Further, Furthermore, we derive the convergence rate of $\sqrt{n}(\tilde{\theta}_{n} - \theta)$ to a normal distribution, that is, the Berry-Esséen type upper bound. This result is superior to the work of Douissi et al. (2022) in the framework of stationary Gaussian processes, as it not only provides a sharper upper bound but also applies to a wide range of stationary and non-stationary fractional Gaussian processes, including fractional Brownian motion, sub-fractional Brownian motion, bi-fractional Brownian motion, generalized sub-fractional Brownian motion, and general fractional Brownian motion. The key method is the formula for estimating the Kolmogorov distance between two random variables based on the Delta method obtained by Tang et al. (2025).

Key words: Ornstein-Uhlenbeck process, fractional Brownian motion, fractional Gaussian process, Berry-Esséen bound