Asymptotic covariances for functionals of weakly stationary random fields

Abstract

Let $(A_x)_{x\in\mathbb{R}^d}$ be a locally integrable, centered, weakly stationary random field, i.e. $\mathbb{E}[A_x]=0$, ${\rm Cov}(A_x,A_y)=K(x-y)$, $\forall x,y\in\mathbb{R}^d$, with measurable covariance function $K:\mathbb{R}^d\rightarrow\mathbb{R}$. Assuming only that $w_t:=\int_{\{|z|\le t\}}K(z)dz$ is regularly varying (which encompasses the classical assumptions found in the literature), we compute $$\lim_{t\rightarrow\infty}{\rm Cov}\left(\frac{\int_{tD}A_x dx}{t^{d/2}w_t^{1/2}}, \frac{\int_{tL}A_y dy}{t^{d/2}w_t^{1/2}}\right)$$ for $D,L\subseteq \mathbb{R}^d$ 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 $K_{\text{iso}}$. 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.