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

¹ Dépt de Mathématiques, Université de Pau, Pau, France
² Cardiff School of Mathematics, Cardiff University, Senghennydd Road, Cardiff, CF24 4AG, UK
³ Dept. of Probability Theory and Mathematical Statistics, Kyiv National Taras Shevchenko University, Ukraine, and Cardiff School of Mathematics, Cardiff University, Senghennydd Road, Cardiff, CF24 4AG, UK

Corresponding authors: Florin.Avram@univ-Pau.fr LeonenkoN@Cardiff.ac.uk lms@univ.kiev.ua

Received: 22 January 2008
Revised: 25 August 2008

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.