Kolmogorov bounds in the CLT of the LSE for Gaussian Ornstein Uhlenbeck processes

Abstract

In this paper, we consider the Ornstein–Uhlenbeck (OU) process defined as solution to the equation [Formula: see text], [Formula: see text], where [Formula: see text] is a Gaussian process with stationary increments, whereas [Formula: see text] is unknown parameter to be estimated. We provide an upper bound in Kolmogorov distance for normal approximation of the least squares estimator [Formula: see text] of the drift parameter [Formula: see text] on the basis of the continuous observation [Formula: see text], as [Formula: see text]. Our approach is based on some novel estimates involving a combination of Malliavin calculus and Stein’s method for normal approximation. We apply our result to fractional OU processes of the first kind, and improved the upper bound of the Kolmogorov distance for the LSE [Formula: see text] provided by [Y. Chen, N. Kuang and Y. Li, Berry–Esseen bound for the parameter estimation of fractional Ornstein–Uhlenbeck processes, Stoch. Dyn. 20(4) (2020) 2050023; Y. Chen and Y. Li, Berry–Esseen bound for the parameter estimation of fractional Ornstein–Uhlenbeck processes with the hurst parameter [Formula: see text], Commun. Stat. Theory Methods 50(13) (2021) 2996–3013], respectively, in the cases [Formula: see text] and [Formula: see text]. We also apply our approach to fractional OU processes of the second kind.