  Many statistical applications require establishing central limit theorems for
sums, integrals, or for quadratic forms of functions of a stationary process. A
particularly important case is that of Appell polynomials, since the Appell
expansion rank" determines typically the type of central limit theorem
satisfied by these functionals. We review and extend here to multidimensional
indices a functional analysis approach to this problem proposed by Avram and
Brown (1989), based on the method of cumulants and on integrability assumptions
in the spectral domain; several applications are presented as well.
