Many statistical applications require establishing central limit theorems for sums/integrals or for quadratic forms , 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.
Mots clés : quadratic forms, Appell polynomials, Hölder-Young inequality, Szegö type limit theorem, asymptotic normality, minimum contrast estimation
@article{PS_2010__14__210_0, author = {Avram, Florin and Leonenko, Nikolai and Sakhno, Ludmila}, title = {On a {Szeg\"o} type limit theorem, the {H\"older-Young-Brascamp-Lieb} inequality, and the asymptotic theory of integrals and quadratic forms of stationary fields}, journal = {ESAIM: Probability and Statistics}, pages = {210--255}, publisher = {EDP-Sciences}, volume = {14}, year = {2010}, doi = {10.1051/ps:2008031}, mrnumber = {2741966}, language = {en}, url = {http://www.numdam.org/articles/10.1051/ps:2008031/} }
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 PY - 2010 SP - 210 EP - 255 VL - 14 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/ps:2008031/ DO - 10.1051/ps:2008031 LA - en ID - PS_2010__14__210_0 ER -
%0 Journal Article %A Avram, Florin %A Leonenko, Nikolai %A Sakhno, Ludmila %T On a Szegö type limit theorem, the Hölder-Young-Brascamp-Lieb inequality, and the asymptotic theory of integrals and quadratic forms of stationary fields %J ESAIM: Probability and Statistics %D 2010 %P 210-255 %V 14 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/ps:2008031/ %R 10.1051/ps:2008031 %G en %F PS_2010__14__210_0
Avram, Florin; Leonenko, Nikolai; 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, Tome 14 (2010), pp. 210-255. doi : 10.1051/ps:2008031. http://www.numdam.org/articles/10.1051/ps:2008031/
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