Berry-Esséen bound for the parameter estimation of fractional Ornstein-Uhlenbeck processes with the hurst parameter H∈(0,12)

Abstract

For an Ornstein–Uhlenbeck process driven by a fractional Brownian motion with Hurst parameter $H\in (0,\frac{1}{2}),$ one shows the Berry–Esséen bound of the least squares estimator of the drift parameter. Thus, a problem left in Chen, Kuang, and Li (2019) is solved, where the Berry–Esséen bound of the least squares estimator is proved for $H\in [\frac{1}{2},\frac{3}{4}].$ A new ingredient is a corollary of the inner product’s representation of the Hilbert space associated with the fractional Brownian motion given by Jolis (2007). An approach based on Malliavin calculus given by Kim and Park (2017b) is used. Several computations are cited from Hu, Nualart, and Zhou (2019).

Keywords:

• Berry–Esséen bound
• Fourth Moment theorems
• fractional Ornstein– Uhlenbeck process
• Malliavin calculus

Acknowledgments

We would like to gratefully thank the referee for very valuable suggestions which lead to the improvement of the new version.

Additional information

Funding

Y. Chen is supported by NSFC (Nos. 11961033, 11871079).