  Statistical modeling for massive spatial data sets has generated a
substantial literature on scalable spatial processes based upon Vecchia's
approximation. Vecchia's approximation for Gaussian process models enables fast
evaluation of the likelihood by restricting dependencies at a location to its
neighbors. We establish inferential properties of microergodic spatial
covariance parameters within the paradigm of fixed-domain asymptotics when they
are estimated using Vecchia's approximation. The conditions required to
formally establish these properties are explored, theoretically and
empirically, and the effectiveness of Vecchia's approximation is further
corroborated from the standpoint of fixed-domain asymptotics.
