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.

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

DOI : 10.1051/ps:2008031
Classification : 60F05, 62M10, 60G15, 62M15, 60G10, 60G60
Mots-clés : quadratic forms, Appell polynomials, Hölder-Young inequality, Szegö type limit theorem, asymptotic normality, minimum contrast estimation 
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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://archive.numdam.org/articles/10.1051/ps:2008031/

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