Anisotropic scaling limits of long-range dependent linear random fields on ${\mathbb{Z}}^3$

Abstract

We provide a complete description of anisotropic scaling limits of stationary linear random field on ${\mathbb{Z}}^3$ with long-range dependence and moving average coefficients decaying as $O(|t_i{|}^{-q_i})$ in the $i$th direction, $i=1,2,3.$ The scaling limits are taken over rectangles in ${\mathbb{Z}}^3$ whose sides increase as $O({\lambda }^{{\gamma }_i}),i=1,2,3$ when $\lambda \to \infty$, for any fixed ${\gamma }_i>0,i=1,2,3$. We prove that all these limits are Gaussian RFs whose covariance structure essentially is determined by the fulfillment or violation of the balance conditions ${\gamma }_i{q}_i={\gamma }_j{q}_j,1\le i<j\le 3$. The paper extends recent results in \cite{ps2015}, \cite{ps2016}, \cite{pils2016}, \cite{pils2017} on anisotropic scaling of long-range dependent random fields from dimension 2 to dimension 3.