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

TL;DR

This paper studies the asymptotic behavior of traces and extreme eigenvalues of products of truncated Toeplitz matrices with singular symbols, providing new results on their limits and implications for large deviation principles.

Contribution

It offers novel asymptotic expansions for traces of Toeplitz matrix products with singular symbols and analyzes extreme eigenvalues, extending understanding in the singular case.

Findings

Asymptotic expansion of trace for specific Toeplitz products.

Limits of extreme eigenvalues as matrix size grows.

Large deviation principles for quadratic forms of stationary processes.

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.