Spatio-temporal models are widely used for inference in statistics and many applied areas. In such contexts interests are often in the geometric nature (e.g. anisotropy), and the statistical properties of these models. This dissertation has two parts. The first part focuses on the sample path properties of space-time models. We apply the theory of Yaglom (1957) to construct a large class of space-time models with stationary increments (also called intrinsically stationary random fields) and study their statistical and geometric properties. We derive upper and lower bounds for the prediction errors, establish criteria for the mean-square and sample path differentiability, all in terms of the parameters of the models explicitly. Moreover, it is shown that when the random fields are not smooth, we can generate various kinds of random fractals and the related Hausdorff dimensions are computed. Our main results show that the statistical and geometric properties of the Gaussian random fields we propose are very different from those obtained by deformation from any isotropic random field; and they can be applied to analyze more general Gaussian intrinsic random functions, convolution-based space-time Gaussian models [Higdon (2002), Calder and Cressie (2007)] and the spatial processes in Fuentes (2002, 2005). The second part of the dissertation pertains to equivalence of Gaussian measures and asymptotically optimal predictions of intrinsically stationary random fields. We extend the methods which Ibragimov and Rozanov (1978) use for stationary processes to study intrinsically stationary random fields. We describe the relationships among three corresponding Hilbert spaces: the random variable space generated by the random field, the corresponding reproducing kernel Hilbert space, and the complex function space spanned by certain analytic functions using the spectral measure. Criteria for equivalence and orthogonality of intrinsically stationary Gaussian random fields are delivered in terms of their spectral measures and the structures of their reproducing kernel Hilbert spaces. Our results are different from those for stationary processes [see Ibragimov and Rozanov (1978)]. Given the equivalence of two Gaussian measures, the asymptotic optimality of linear predictions of intrinsically stationary random fields and the convergence rates are established in this part. Moreover, the asymptotic efficient prediction of non-stationary, anisotropic space-time models with a misspecified probability distribution is studied. The main results show that under the equivalence of two Gaussian measures, the prediction based on the incorrect distribution is asymptotically optimal and efficient relative to the prediction under the correct distribution, as the points of observations become increasingly dense in the study domain. Our results extend those of Stein (1988, 1990, 1999a, 1999b) which were concerned with isotropic and stationary Gaussian random fields.
Read
