On the Kolmogorov Distance for the Least Squares Estimator in the Fractional Ornstein-Uhlenbeck Process

Abstract

The paper shows that the distribution of the normalized least squares estimator of the drift parameter in the fractional Ornstein-Uhlenbeck process observed over [0, T] converges to the standard normal distribution with an uniform optimal error bound of the order O(T ^−1/2) for 0.5 ≤ H ≤ 0.63 and of the order O(T^4H-3) for 0.63 < H < 0.75 where H is the Hurst exponent of the fractional Brownian motion driving the Ornstein-Uhlenbeck process. For the normalized quasi-least squares estimator, the error bound is of the order O(T^−1/4) for 0.5 ≤ H ≤ 0.69 and of the order O(T^4H−3) for 0.69 < H < 0.75.