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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 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.

Keywords

Quadratic formsAppell polynomialsHölder-YounginequalitySzegö type limit theoremasymptotic normalityminimum contrast estimation

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Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 14 , 2010 , pp. 210 - 255
Copyright
© EDP Sciences, SMAI, 2010

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