Convergence of quadratic forms for random fields and its application to the convergence of empirical covariances

Frédéric Lavancier (1) , Anne Philippe (1)

1 LMJL - Laboratoire de Mathématiques Jean Leray

Limit theorems are proved for quadratic forms of Gaussian random fields in presence of long memory. Similarly to the one dimensional case, we prove that the quadratic forms, appropriately normalized, may have Gaussian or non-Gaussian limits. However the dichotomy observed in $d=1$ cannot be stated so easily, due to the possible occurrence of anisotropic strong dependence in $d>1$. We apply our theorems to obtain the asymptotic behavior of the empirical covariances, which is a particular example of quadratic forms.

Mots clés: Gaussian Random field; Long memory; Central and non central limit theorems

Domaines: Statistiques [math.ST]

Frédéric Lavancier, Anne Philippe. Convergence of quadratic forms for random fields and its application to the convergence of empirical covariances. 2008. ⟨hal-00348833v1⟩