  Vecchia approximation has been widely used to accurately scale
Gaussian-process (GP) inference to large datasets, by expressing the joint
density as a product of conditional densities with small conditioning sets. We
study fixed-domain asymptotic properties of Vecchia-based GP inference for a
large class of covariance functions (including Mat\'ern covariances) with
boundary conditioning. In this setting, we establish that consistency and
asymptotic normality of maximum exact-likelihood estimators imply those of
maximum Vecchia-likelihood estimators, and that exact GP prediction can be
approximated accurately by Vecchia GP prediction, given that the size of
conditioning sets grows polylogarithmically with the data size. Hence,
Vecchia-based inference with quasilinear complexity is asymptotically
equivalent to exact GP inference with cubic complexity. This also provides a
general new result on the screening effect. Our findings are illustrated by
numerical experiments, which also show that Vecchia approximation can be more
accurate than alternative approaches such as covariance tapering and
reduced-rank approximations.
