  Covariance parameter estimation of Gaussian processes is analyzed in an
asymptotic framework. The spatial sampling is a randomly perturbed regular grid
and its deviation from the perfect regular grid is controlled by a single
scalar regularity parameter. Consistency and asymptotic normality are proved
for the Maximum Likelihood and Cross Validation estimators of the covariance
parameters. The asymptotic covariance matrices of the covariance parameter
estimators are deterministic functions of the regularity parameter. By means of
an exhaustive study of the asymptotic covariance matrices, it is shown that the
estimation is improved when the regular grid is strongly perturbed. Hence, an
asymptotic confirmation is given to the commonly admitted fact that using
groups of observation points with small spacing is beneficial to covariance
function estimation. Finally, the prediction error, using a consistent
estimator of the covariance parameters, is analyzed in details.
