  We deal with the comparison of space-time covariance kernels having, either,
full, spatially dynamical, or space-time compact support. Such a comparison is
based on compatibility of these covariance models under fixed domain
asymptotics, having a theoretical background that is substantially coming from
equivalence or orthogonality of Gaussian measures. In turn, such a theory is
intimately related to the tails of the spectral densities associated with the
three models. Models with space-time compact support are still elusive. We
taper the temporal part of a model with dynamical support, obtaining a
space-time compact support. The spectrum related to such a construction is
obtained through temporal convolution of the spatially dynamical spectrum with
the spectrum associated with the temporal taper. The solution of such a
challenge opens the door to the compatibility-based comparison. Our findings
show that indeed these three models can be compatible under some suitable
parametric restrictions. As a corollary, we deduce implications in terms of
maximum likelihood estimation and misspecified kriging prediction under fixed
domain asymptotics.
