Abstract: We study the estimation and prediction of Gaussian processes with space-time covariance models belonging to the dynamical generalized Wendland (𝒟𝒢𝒲) family, under fixed-domain asymptotics. Such a class is nonseparable, has dynamical compact supports, and parameterizes differentiability at the origin similarly to the space-time Matérn class. Our results are presented in two parts. First, we establish the strong consistency and asymptotic normality for the maximum likelihood estimator of the microergodic parameter associated with the 𝒟𝒢𝒲 covariance model, under fixed-domain asymptotics. The second part focuses on optimal kriging prediction under the 𝒟𝒢𝒲 model and an asymptotically correct estimation of the mean squared error using a misspecified model. Our theoretical results are, in turn, based on the equivalence of Gaussian measures under some given families of space-time covariance functions, where both space or time are compact. The technical results are provided in the online Supplementary material.

Key words and phrases: Fixed-domain asymptotics, microergodic parameter, maximum likelihood, space-time generalized wendland family.