  In geostatistical problems with massive sample size, Gaussian processes can
be approximated using sparse directed acyclic graphs to achieve scalable $O(n)$
computational complexity. In these models, data at each location are typically
assumed conditionally dependent on a small set of parents which usually include
a subset of the nearest neighbors. These methodologies often exhibit excellent
empirical performance, but the lack of theoretical validation leads to unclear
guidance in specifying the underlying graphical model and sensitivity to graph
choice. We address these issues by introducing radial neighbors Gaussian
processes (RadGP), a class of Gaussian processes based on directed acyclic
graphs in which directed edges connect every location to all of its neighbors
within a predetermined radius. We prove that any radial neighbors Gaussian
process can accurately approximate the corresponding unrestricted Gaussian
process in Wasserstein-2 distance, with an error rate determined by the
approximation radius, the spatial covariance function, and the spatial
dispersion of samples. We offer further empirical validation of our approach
via applications on simulated and real world data showing excellent performance
in both prior and posterior approximations to the original Gaussian process.
