  Gaussian process models typically contain finite dimensional parameters in
the covariance function that need to be estimated from the data. We study the
Bayesian fixed-domain asymptotics for the covariance parameters in a universal
kriging model with an isotropic Matern covariance function, which has many
applications in spatial statistics. We show that when the dimension of domain
is less than or equal to three, the joint posterior distribution of the
microergodic parameter and the range parameter can be factored independently
into the product of their marginal posteriors under fixed-domain asymptotics.
The posterior of the microergodic parameter is asymptotically close in total
variation distance to a normal distribution with shrinking variance, while the
posterior distribution of the range parameter does not converge to any point
mass distribution in general. Our theory allows an unbounded prior support for
the range parameter and flexible designs of sampling points. We further study
the asymptotic efficiency and convergence rates in posterior prediction for the
Bayesian kriging predictor with covariance parameters randomly drawn from their
posterior distribution. In the special case of one-dimensional
Ornstein-Uhlenbeck process, we derive explicitly the limiting posterior of the
range parameter and the posterior convergence rate for asymptotic efficiency in
posterior prediction. We verify these asymptotic results in numerical
experiments.
