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

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 $d>1$.