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

Abstract

Let Z:={Zt,t≥0} be a stationary Gaussian process. We study two estimators of E[Z02], namely fˆT(Z):=1 T ∫0TZt2dt, and f˜n(Z):=1 n ∑i=1nZti2, where ti=iΔn, i=0,1,…,n, Δn→0 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.

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