Many statistical applications require establishing
central limit theorems for sums/integrals
$S_T(h)=\int_{t \in I_T} h (X_t) {\rm d}t$ or for quadratic forms $Q_T(h)=\int_{t,s \in I_T} \hat{b}(t-s) h (X_t, X_s) {\rm d}s {\rm d}t$, where Xt is a stationary
process. A particularly important case is that of Appell
polynomials h(Xt) = Pm(Xt), h(Xt,Xs) = Pm,n (Xt,Xs), since the “Appell expansion rank" determines typically the
type of central limit theorem satisfied by the functionals
ST(h), QT(h).
We review and extend here to multidimensional
indices, along lines conjectured in [F. Avram and M.S. Taqqu, Lect. Notes Statist.187 (2006) 259–286], a functional
analysis approach to this problem proposed by [Avram and Brown, Proc. Amer.
Math. Soc.107 (1989) 687–695] based on the method of cumulants and on integrability
assumptions in the spectral domain; several applications are
presented as well.