Berry–Esséen bounds for the least squares estimator for discretely observed fractional Ornstein–Uhlenbeck processes

Abstract

Let θ>0. We consider a one-dimensional fractional Ornstein–Uhlenbeck process defined as dXt=−θXtdt+dBt,t≥0, where B is a fractional Brownian motion of Hurst parameter H∈(12,1). We are interested in the problem of estimating the unknown parameter θ. For that purpose, we dispose of a discretized trajectory, observed at n equidistant times ti=iΔn,i=0,…,n, and Tn=nΔn denotes the length of the ‘observation window’. We assume that Δn→0 and Tn→∞ as n→∞. As an estimator of θ we choose the least squares estimator (LSE) θ̂n. The consistency of this estimator is established. Explicit bounds for the Kolmogorov distance, in the case when H∈(12,34), in the central limit theorem for the LSE θ̂n are obtained. These results hold without any kind of ergodicity on the process X.