Trace et valeurs propres extrêmes 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 (θ) = |1 - e^{i θ} | ^{2{α1}c1 (eiθθ) and f2 (θ) = |1 - e ^{iθ}| ^{2α2}c2 (eiθ), with c1 and c2 are two regular functions of the torus and - 1/2 < α1, α2 < 1/2 . In a second part of this work we study the particular case where α1 > 0 and α2 < 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.