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

TL;DR

This paper characterizes the anisotropic scaling limits of long-range dependent linear random fields on three-dimensional integer lattices, revealing Gaussian limits determined by balance conditions among decay rates and scaling exponents.

Contribution

It extends previous two-dimensional results to three dimensions, providing a complete description of Gaussian scaling limits for anisotropic long-range dependent fields.

Findings

Scaling limits are Gaussian random fields.

Balance conditions determine the covariance structure.

Results generalize previous 2D findings to 3D.

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.