Abstract: The Matérn class is an important class of covariance functions in spatial statistics. With the recent flourishing trend in modelling spatio-temporal data, indepth theoretical development of spatio-temporal covariograms is needed. In this paper, theories under the infill asymptotic framework concerning estimation issues of a generally non-separable Matérn class of spatio-temporal covariance function is presented. It is found that not all parameters can be estimated consistently while quantities that can be estimated consistently are found based on equivalence and orthogonality of Gaussian measures. The micro-ergodic parameters are found to be different when the degrees of separability between the space and time components are different. For the computation, an easy-to-implement estimation procedure is given. Simulation studies are conducted to show how well the asymptotic results apply when the sample size is moderate. A set of air pollution data is used to demonstrate the usefulness of the estimation procedure suggested.

Key words and phrases: Gaussian measures, infill asymptotics, micro-ergodic parameters, space-time data.