Spatial circulants, with applications

The cumulants of quadratic forms associated to the so-called spatial design matrices are often needed for inference in the context of isotropic processes on uniform grids. Because the eigenvalues of the matrices involved are generally unknown, the computation of the cumulants can be very demanding if the grids are large. This paper first replaces the spatial design matrices with circular counterparts having known eigenvalues. It then studies some of the properties of the approximating matrices, and analyzes their performance in a number of applications to well-known inferential procedures.