  Let $Z:=\{Z_t,t\geq0\}$ be a stationary Gaussian process. We study two
estimators of $\mathbb{E}[Z_0^2]$, namely $\widehat{f}_T(Z):= \frac{1}{T}
\int_{0}^{T} Z_{t}^{2}dt$, and $\widetilde{f}_n(Z) :=\frac{1}{n} \sum_{i
=1}^{n} Z_{t_{i}}^{2}$, where $ t_{i} = i \Delta_{n}$, $ i=0,1,\ldots, n $,
$\Delta_{n}\rightarrow 0$ and $ T_{n} := n \Delta_{n}\rightarrow \infty$. We
prove that the two estimators are strongly consistent and establish
Berry-Esseen bounds for a central limit theorem involving $\widehat{f}_T(Z)$
and $\widetilde{f}_n(Z)$. We apply these results to asymptotically stationary
Gaussian processes and estimate the drift parameter for Gaussian
Ornstein-Uhlenbeck processes.
