  We consider the class of all stationary Gaussian process with explicit
parametric spectral density. Under some conditions on the autocovariance
function, we defined a GMM estimator that satisfies consistency and asymptotic
normality, using the Breuer-Major theorem and previous results on ergodicity.
This result is applied to the joint estimation of the three parameters of a
stationary Ornstein-Uhlenbeck (fOU) process driven by a fractional Brownian
motion. The asymptotic normality of its GMM estimator applies for any H in
(0,1) and under some restrictions on the remaining parameters. A numerical
study is performed in the fOU case, to illustrate the estimator's practical
performance when the number of datapoints is moderate.
