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

Let Z := {Z(t) ,t >= 0} be a stationary Gaussian process. We study two estimators of E[Z(0)(2)], namely (f) over cap (T)(Z) := 1/T integral(T)(0) Z(t)(2)dt, and (f) over tilde (n) (Z) := 1/n Sigma(n)(i=1) Z(ti)(2), where t(i) = i Delta(n), i = 0, 1, ..., n, Delta -> 0 and T-n := n Delta(n) -> infinity. We prove that the two estimators are strongly consistent and establish Berry-Esseen bounds for a central limit theorem involving (f) over cap (T)(Z) and (f) over tilde (n)(Z). We apply these results to asymptotically stationary Gaussian processes and estimate the drift parameter for Gaussian Ornstein-Uhlenbeck processes.