Berry-Ess\'een bounds for the least squares estimator for discretely observed fractional Ornstein-Uhlenbeck processes

TL;DR

This paper derives Berry-Esséen bounds for the least squares estimator of the drift parameter in a fractional Ornstein-Uhlenbeck process observed discretely, providing explicit convergence rates in the non-ergodic case.

Contribution

It establishes the first explicit Berry-Esséen bounds for the LSE of the fractional Ornstein-Uhlenbeck process with Hurst parameter in (1/2, 3/4), without assuming ergodicity.

Findings

Established consistency of the LSE.

Derived explicit Kolmogorov distance bounds in the CLT.

Results hold for non-ergodic fractional Ornstein-Uhlenbeck processes.

Abstract

Let $\theta>0$. We consider a one-dimensional fractional Ornstein-Uhlenbeck process defined as $dX_t= -\theta\ X_t dt+dB_t,\quad t\geq0,$ where $B$ is a fractional Brownian motion of Hurst parameter $H\in(1/2,1)$. We are interested in the problem of estimating the unknown parameter $\theta$. For that purpose, we dispose of a discretized trajectory, observed at $n$ equidistant times $t_i=i\Delta_{n}, i=0,...,n$, and $T_n=n\Delta_{n}$ denotes the length of the `observation window'. We assume that $\Delta_{n} \rightarrow 0$ and $T_n\rightarrow \infty$ as $n\rightarrow \infty$. As an estimator of $\theta$ we choose the least squares estimator (LSE) $\hat{\theta}_{n}$. The consistency of this estimator is established. Explicit bounds for the Kolmogorov distance, in the case when $H\in(1/2,3/4)$, in the central limit theorem for the LSE $\hat{\theta}_{n}$ are obtained. These results hold without any kind of ergodicity on the process $X$.