Optimal Berry-Esseen bound for an estimator of parameter in the Ornstein-Uhlenbeck process

This paper is concerned with the study of the rate of central limit theorem for the maximum likelihood estimator θˆT of the unknown parameter θ>0, based on the observation X={Xt,0≤t≤T}, occurring in the drift coefficient of an Ornstein-Uhlenbeck process dXt=−θXtdt+dWt,X0=0 for 0≤t≤T, where {Wt,t≥0} is a standard Brownian motion. The tool we use is an Edgeworth expansion with an explicitly expressed remainder. We prove that upper and lower bounds, obtained by controlling the remainder term, give an optimal rate 1T in Kolmogorov distance for normal approximation of θˆT.