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 or for quadratic forms , 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.