On a Szegö type limit theorem, the Hölder-Young-Brascamp-Lieb inequality, and the asymptotic theory of integrals and quadratic forms of stationary fields

Abstract

Many statistical applications require establishing central limit theorems for sums/integrals ST(h)=∫tεIT} h (Xt) dt or for quadratic forms QT(h)=∫t,sεIT b̂(t-s) h(Xt, Xs) dsdt, 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. © EDP Sciences, SMAI, 2010.