This paper considers the convergence in distribution of quadratic forms defined as the weighted sum of the cross-products from a sequence of $d$-dimensional stationary $L^2$ Gaussian random fields having long memory. When $d=1$, both central and non-central limit theorems have been proved under various conditions on the spectral density $f$ of the sequence and weight function $g$ defining the quadratic forms. When $d>1$, a central limit theorem has been proved but few results are available on a non-central limit theorem. In this paper, a non-central limit theorem is provided for the convergence of the quadratic form under a general condition on $f$ and $g$ in the case $d>1$. Some examples, where the general condition is satisfied, are given. As an application of the main result, the asymptotic distributions of an empirical auto-covariance function are derived under various conditions.
Reviewer: Dongsheng Tu (Kingston)