Volume 14, 2010
|Page(s)||210 - 255|
|Published online||29 July 2010|
On a Szegö type limit theorem, the Hölder-Young-Brascamp-Lieb inequality, and the asymptotic theory of integrals and quadratic forms of stationary fields *
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
Revised: 25 August 2008
Many statistical applications require establishing central limit theorems for sums/integrals or for quadratic forms , 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.
Mathematics Subject Classification: 60F05 / 62M10 / 60G15 / 62M15 / 60G10 / 60G60
Key words: Quadratic forms / Appell polynomials / Hölder-Young inequality / Szegö type limit theorem / asymptotic normality / minimum contrast estimation
© EDP Sciences, SMAI, 2010
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