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

Author： Avram Florin Leonenko Nikolai Sakhno Ludmila

Publisher： Edp Sciences

E-ISSN： 1262-3318|14|issue|210-255

ISSN： 1292-8100

Source： ESAIM: Probability and Statistics, Vol.14, Iss.issue, 2010-07, pp. : 210-255

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Abstract

Many statistical applications require establishingcentral 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 X_t is a stationaryprocess. A particularly important case is that of Appellpolynomials 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 thetype of central limit theorem satisfied by the functionalsS_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 functionalanalysis 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 integrabilityassumptions in the spectral domain; several applications arepresented as well.