Trace et valeurs propres extr\^emes d'un produit de matrices de Toeplitz. Le cas singulier

Abstract

Trace and extreme eigenvalues of a product of truncated Toeplitz matrices. The singular case. In a first theorem we give an asymptotic expansion of Tr (T_N (f_1) T_N^{-1}(f_2)) where f1 ({\theta}) = |1 - e^{i {\theta}} | ^{2{{\alpha}1}c1 (ei{\theta}{\theta}) and f2 ({\theta}) = |1 - e ^{i{\theta}}| ^{2{\alpha}2}c2 (ei{\theta}), with c1 and c2 are two regular functions of the torus and - 1/2 < {\alpha}1, {\alpha}2 < 1/2 . In a second part of this work we study the particular case where {\alpha}1 > 0 and {\alpha}2 < 0. Then we obtain the asymptotic of the trace of the powers of Tr (T_N (f_1) T_N^{-1}(f_2)) for s {\in} N* that provides us the limits when N goes to the infinity of the extreme eigenvalues of this matrix. This last result allows us to give a large deviation principle for a family of quadratic forms of stationnary process.