A note on approximations of traces of products of truncated Toeplitz matrices

The paper establishes error orders for integral limit approximations to the traces of products of truncated Toeplitz matrices generated by integrable real symmetric functions defined on the unit circle. These approximations and the corresponding error bounds are of importance in the statistical analysis of discrete-time stationary processes (asymptotic distributions and large deviations of Toeplitz type quadratic forms, estimation of the spectral parameters and functionals, asymptotic expansions of the estimators, etc.). The results improve the rates obtained by the authors in an earlier paper.