Berry-Esseen bounds of second moment estimators for Gaussian processes observed at high frequency

Abstract

Let $Z:=\{Z_t,t\ge 0\}$ be a stationary Gaussian process. We study two estimators of $\mathbb{E}[Z_0^2]$, namely ${\hat{f}}_T(Z):=\frac{1}{T}{\int }_0^T{Z}_t^{2}dt$, and ${\tilde{f}}_n(Z):=\frac{1}{n}{\sum }_{i=1}^n{Z}_{t_i}^2$, where $t_i=i{\mathrm{\Delta }}_n$, $i=0,1,\dots ,n$, ${\mathrm{\Delta }}_n\to 0$ and $T_n:=n{\mathrm{\Delta }}_n\to \mathrm{\infty }$. We prove that the two estimators are strongly consistent and establish Berry-Esseen bounds for a central limit theorem involving ${\hat{f}}_T(Z)$ and ${\tilde{f}}_n(Z)$. We apply these results to asymptotically stationary Gaussian processes and estimate the drift parameter for Gaussian Ornstein-Uhlenbeck processes.