On a Szego Type Limit Theorem, the Holder-young-brascamp-lieb Inequality, and the Asymptotic Theory of Integrals and Quadratic Forms of Stationary Fields

Many statistical applications require establishing central limit theorems for sums/integrals ST (h) = ∫ t∈IT h(Xt)dt or for quadratic forms QT (h) = ∫ t,s∈IT 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 [16], a functional analysis approach to this problem proposed by Avram and Brown (1989), based on the method of cumulants and on integrability assumptions in the spectral domain; several applications are presented as well. Résumé. Nous considérons ici des théorémes de limite central pour des sommes/integrales ST (h) = ∫ t∈IT h(Xt)dt, et pour des formes quadratiques QT (h) = ∫ t,s∈IT b̂(t − s)h(Xt,Xs)dsdt, où Xt est un processus stationnaire. Un cas particulièrement important est celui des polynômes d’Appell h(Xt) = Pm(Xt), h(Xt,Xs) = Pm,n(Xt, Xs). Pour ce problème, nous généralisons ici au cas des indices multidimensionnels une approche proposée par Avram et Brown (1989), basée sur la méthode des cumulants et sur des hypothèses d’integrabilité dans le domaine spectral. Plusieurs applications illustrent la versatilité de l’approche. 1991 Mathematics Subject Classification. 60F05, 62M10, 60G15, 62M15, 60G10, 60G60. The dates will be set by the publisher.