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 X
t
is a stationary
process. A particularly important case is that of Appell
polynomials h(X
t
) = P
m
(X
t
), h(X
t
,X
s
) = P
m
,
n
(X
t
,X
s
), since the “Appell expansion rank" determines typically the
type of central limit theorem satisfied by the functionals
S
T
(h), Q
T
(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.