Limit theorems for anisotropic functionals of stationary Gaussian fields with Gneiting covariance function

TL;DR

This paper establishes limit theorems for non-linear functionals of stationary Gaussian fields with Gneiting covariance, revealing conditions for Gaussian or non-Gaussian limits in anisotropic, long-range dependent settings.

Contribution

It provides the first comprehensive analysis of limit distributions for anisotropic Gaussian fields with Gneiting covariance, including explicit conditions and structural insights.

Findings

Identifies regimes for Gaussian and Rosenblatt limit distributions

Shows asymptotic separability of Gneiting covariances in cumulant sense

Extends limit theorems to non-separable, long-range dependent spatiotemporal models

Abstract

We study non-linear additive functionals of stationary Gaussian fields over anisotropically growing domains in $\mathbb{R}^d$, including spatiotemporal settings, and establish Gaussian and non-Gaussian limit theorems under non-separable covariance structures. We characterize the regimes in which the normalized functionals converge either to a Gaussian distribution or to a $2$-domain Rosenblatt distribution, depending on precise long-range dependence conditions. Our analysis covers covariance functions from the Gneiting class, which provides a canonical family of non-separable spatiotemporal models. A key structural result shows that such covariances are asymptotically separable in a precise cumulant sense, allowing us to identify explicitly the limiting distributions without imposing additional spectral assumptions. These results extend existing spatiotemporal limit theorems beyond separable and short-memory frameworks and provide a unified description of anisotropic long-range dependence phenomena.