# 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

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topAvram, 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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