  The Karhunen-Lo\`eve expansion and the Fredholm determinant formula are used
to derive an asymptotic Rosenblatt-type distribution of a sequence of integrals
of quadratic functions of Gaussian stationary random fields on R^d displaying
long-range dependence. This distribution reduces to the usual Rosenblatt
distribution when d=1. Several properties of this new distribution are
obtained. Specifically, its series representation in terms of independent
chi-squared random variables is given, the asymptotic behavior of the
eigenvalues, its L\`evy-Khintchine representation, as well as its membership to
the Thorin subclass of self-decomposable distributions. The existence and
boundedness of its probability density is then a direct consequence.
