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

Many statistical applications require establishing central limit theorems for sums/integrals or for quadratic forms , where is a stationary process. A particularly important case is that of Appell polynomials () = (), (,) = (,), since the "Appell expansion rank" determines typically the type of central limit theorem satisfied by the functionals (), (). We review and extend here to multidimensional indices, along lines conjectured in [F. Avram and M.S. Taqqu, (2006) 259-286], a functional analysis approach to this problem proposed by [Avram and Brown, (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

Florin Avram, Nikolai Leonenko, Ludmila Sakhno. 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, 2010, 14, pp.210-255. ⟨10.1051/ps:2008031⟩. ⟨hal-00612383⟩