Open Access
2022 Berry-Esseen bounds of second moment estimators for Gaussian processes observed at high frequency
Soukaina Douissi, Khalifa Es-Sebaiy, George Kerchev, Ivan Nourdin
Author Affiliations +
Electron. J. Statist. 16(1): 636-670 (2022). DOI: 10.1214/21-EJS1967


Let Z:={Zt,t0} be a stationary Gaussian process. We study two estimators of E[Z02], namely fˆT(Z):=1T0TZt2dt, and f˜n(Z):=1ni=1nZti2, where ti=iΔn, i=0,1,,n, Δn0 and Tn:=nΔn. We prove that the two estimators are strongly consistent and establish Berry-Esseen bounds for a central limit theorem involving fˆT(Z) and f˜n(Z). We apply these results to asymptotically stationary Gaussian processes and estimate the drift parameter for Gaussian Ornstein-Uhlenbeck processes.

Funding Statement

G. Kerchev and I. Nourdin were supported by the FNR grant APOGEe (R-AGR-3585-10) at Luxembourg University.


We would like to thank the referee for his/her careful readings, constructive remarks and useful suggestions.


Download Citation

Soukaina Douissi. Khalifa Es-Sebaiy. George Kerchev. Ivan Nourdin. "Berry-Esseen bounds of second moment estimators for Gaussian processes observed at high frequency." Electron. J. Statist. 16 (1) 636 - 670, 2022.


Received: 1 February 2021; Published: 2022
First available in Project Euclid: 10 January 2022

arXiv: 2102.04810
MathSciNet: MR4361751
zbMATH: 1487.60050
Digital Object Identifier: 10.1214/21-EJS1967

Primary: 60F05
Secondary: 60G10 , 60G15 , 62F12 , 62M09

Keywords: continuous-time observation , High frequency data , Parameter estimation , rate of normal convergence of the estimators , Stationary Gaussian processes , strong consistency

Vol.16 • No. 1 • 2022
Back to Top