Berry-Ess\'een bound for drift estimation of fractional Ornstein Uhlenbeck process of second kind

TL;DR

This paper establishes a Berry-Esséen bound of order 1/√T for the normal approximation of the least squares estimator of the drift parameter in a fractional Ornstein-Uhlenbeck process of the second kind, using Malliavin calculus and Stein's method.

Contribution

It provides the first explicit Berry-Esséen bound for the drift estimator in this specific fractional Ornstein-Uhlenbeck process of the second kind.

Findings

Achieved an O(1/√T) bound in Kolmogorov distance for the estimator's normal approximation.

Demonstrated the effectiveness of Malliavin calculus combined with Stein's method in this context.

Extended the understanding of asymptotic distributional properties of drift estimators in fractional processes.

Abstract

In the present paper we consider the Ornstein-Uhlenbeck process of the second kind defined as solution to the equation $dX_{t} = -\alpha X_{t}dt+dY_{t}^{(1)}, \ \ X_{0}=0$, where $Y_{t}^{(1)}:=\int_{0}^{t}e^{-s}dB^H_{a_{s}}$ with $a_{t}=He^{\frac{t}{H}}$, and $B^H$ is a fractional Brownian motion with Hurst parameter $H\in(\frac12,1)$, whereas $\alpha>0$ is unknown parameter to be estimated. We obtain the upper bound $O(1/\sqrt{T})$ in Kolmogorov distance for normal approximation of the least squares estimator of the drift parameter $\alpha$ on the basis of the continuous observation $\{X_t,t\in[0,T]\}$, as $T\rightarrow\infty$. Our method is based on the work of \cite{kp-JVA}, which is proved using a combination of Malliavin calculus and Stein's method for normal approximation.