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On a Szegö type limit theorem, the Hölder-Young-Brascamp-Liebinequality, and the asymptotic theory ofintegrals and quadratic forms of stationary fields *

Published online by Cambridge University Press:  29 July 2010

Florin Avram ,
Nikolai Leonenko  and
Ludmila Sakhno
Florin Avram
Affiliation:
Dépt de Mathématiques, Université de Pau, Pau, France
Nikolai Leonenko
Affiliation:
Cardiff School of Mathematics, Cardiff University, Senghennydd Road, Cardiff, CF24 4AG, UK
Ludmila Sakhno
Affiliation:
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
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Abstract

Many statistical applications require establishing central limit theorems for sums/integrals $S_T(h)=\int_{t \in I_T} h (X_t) {\rm d}t$ or for quadratic forms $Q_T(h)=\int_{t,s \in I_T} \hat{b}(t-s) h (X_t, X_s) {\rm d}s {\rm d}t$, 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.

Keywords

Quadratic formsAppell polynomialsHölder-Young inequalitySzegö type limit theoremasymptotic normalityminimum contrast estimation
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 14 , 2010 , pp. 210 - 255
Copyright
© EDP Sciences, SMAI, 2010

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