Asymptotic covariances for functionals of weakly stationary random fields

Let (Ax)x∈R be a measurable, weakly stationary random field, i.e. E[Ax]=E[Ay], Cov(Ax,Ay)=K(x−y), ∀x,y∈Rd, with covariance function K:Rd→R. Assuming only that the integral covariance function wt≔∫{|z|≤t}K(z)dz is regularly varying (which encompasses the classical assumptions found in the literature), we compute [Formula presented] for D,L⊆Rd belonging to a certain class of compact sets. As an application, we combine this result with existing limit theorems to obtain multi-dimensional limit theorems for non-linear functionals of stationary Gaussian fields, in particular proving new results for the Berry's random wave model. At the end of the paper, we also show how the problem for A with a general continuous covariance function K can be reduced to the same problem for a radial, continuous covariance function Kiso. The novel ideas of this work are mainly based on regularity conditions for (cross) covariograms of Euclidean sets and standard properties of regularly varying functions.