Some convergence results on quadratic forms for random fields and application to empirical covariances

TL;DR

This paper establishes limit theorems for quadratic forms of Gaussian random fields with long memory, revealing conditions under which empirical covariances converge to Gaussian limits even with non-square-integrable spectral densities.

Contribution

It extends limit theorems for quadratic forms to anisotropic Gaussian fields with long-range dependence, including cases where spectral density is not square-integrable.

Findings

Non-central limit theorem under minimal conditions

Gaussian limits possible without spectral density in L^2

Anisotropic dependence affects convergence behavior

Abstract

Limit theorems are proved for quadratic forms of Gaussian random fields in presence of long memory. We obtain a non central limit theorem under a minimal integrability condition, which allows isotropic and anisotropic models. We apply our limit theorems and those of Ginovian (99) to obtain the asymptotic behavior of the empirical covariances of Gaussian fields, which is a particular example of quadratic forms. We show that it is possible to obtain a Gaussian limit when the spectral density is not in $L^2$. Therefore the dichotomy observed in dimension $d=1$ between central and non central limit theorems cannot be stated so easily due to possible anisotropic strong dependence in $d>1$.