  The asymptotic analysis of covariance parameter estimation of Gaussian
processes has been subject to intensive investigation. However, this asymptotic
analysis is very scarce for non-Gaussian processes. In this paper, we study a
class of non-Gaussian processes obtained by regular non-linear transformations
of Gaussian processes. We provide the increasing-domain asymptotic properties
of the (Gaussian) maximum likelihood and cross validation estimators of the
covariance parameters of a non-Gaussian process of this class. We show that
these estimators are consistent and asymptotically normal, although they are
defined as if the process was Gaussian. They do not need to model or estimate
the non-linear transformation. Our results can thus be interpreted as a
robustness of (Gaussian) maximum likelihood and cross validation towards
non-Gaussianity. Our proofs rely on two technical results that are of
independent interest for the increasing-domain asymptotic literature of spatial
processes. First, we show that, under mild assumptions, coefficients of
inverses of large covariance matrices decay at an inverse polynomial rate as a
function of the corresponding observation location distances. Second, we
provide a general central limit theorem for quadratic forms obtained from
transformed Gaussian processes. Finally, our asymptotic results are illustrated
by numerical simulations.
