Abstract: Statistical modeling for massive spatial data sets has generated a substantial body of literature on scalable spatial processes based on 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 the parameters are estimated using Vecchia's approximation. We explore the conditions required to formally establish these properties, theoretically and empirically. Our results further corroborate the effectiveness of Vecchia's approximation from the standpoint of fixed-domain asymptotics.

Key words and phrases: Fixed-domain asymptotics, Gaussian processes, Matérn covariance function, microergodic parameters, spatial statistics.