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Many statistical applications require establishing central limit theorems forsums, integrals, or for quadratic forms of functions of a stationary process. Aparticularly important case is that of Appell polynomials, since the Appellexpansion rank" determines typically the type of central limit theoremsatisfied by these functionals. We review and extend here to multidimensionalindices a functional analysis approach to this problem proposed by Avram andBrown (1989), based on the method of cumulants and on integrability assumptionsin the spectral domain; several applications are presented as well.

On a Szego Type Limit Theorem, the Holder-Young-Brascamp-Lieb  Inequality, and the Asymptotic Theory of Integrals and Quadratic Forms of  Stationary Fields