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

Florin Avram; Nikolai Leonenko; Ludmila Sakhno

ESAIM: Probability and Statistics (2010)

  • Volume: 14, page 210-255
  • ISSN: 1292-8100

Abstract

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Many statistical applications require establishing central limit theorems for sums/integrals S T ( h ) = ∫ t ∈ I T h ( X t ) d t or for quadratic forms Q T ( h ) = ∫ t , s ∈ I T b ^ ( t - s ) h ( X t , X s ) d s 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.

How to cite

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Avram, Florin, Leonenko, Nikolai, and Sakhno, Ludmila. "On a Szegö type limit theorem, the Hölder-Young-Brascamp-Lieb inequality, and the asymptotic theory of integrals and quadratic forms of stationary fields *." ESAIM: Probability and Statistics 14 (2010): 210-255. <http://eudml.org/doc/250860>.

@article{Avram2010,
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. },
author = {Avram, Florin, Leonenko, Nikolai, Sakhno, Ludmila},
journal = {ESAIM: Probability and Statistics},
keywords = {Quadratic forms; Appell polynomials; Hölder-Young inequality; Szegö type limit theorem; asymptotic normality; minimum contrast estimation; Hölder-Young inequality; Fejér graph integrals},
language = {eng},
month = {7},
pages = {210-255},
publisher = {EDP Sciences},
title = {On a Szegö type limit theorem, the Hölder-Young-Brascamp-Lieb inequality, and the asymptotic theory of integrals and quadratic forms of stationary fields *},
url = {http://eudml.org/doc/250860},
volume = {14},
year = {2010},
}

TY - JOUR
AU - Avram, Florin
AU - Leonenko, Nikolai
AU - Sakhno, Ludmila
TI - On a Szegö type limit theorem, the Hölder-Young-Brascamp-Lieb inequality, and the asymptotic theory of integrals and quadratic forms of stationary fields *
JO - ESAIM: Probability and Statistics
DA - 2010/7//
PB - EDP Sciences
VL - 14
SP - 210
EP - 255
AB - 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.
LA - eng
KW - Quadratic forms; Appell polynomials; Hölder-Young inequality; Szegö type limit theorem; asymptotic normality; minimum contrast estimation; Hölder-Young inequality; Fejér graph integrals
UR - http://eudml.org/doc/250860
ER -

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