Error Bounds and Asymptotic Expansions for Toeplitz Product Functionals of Unbounded Spectra∗

This paper establishes error orders for integral limit approximations to traces of powers (to the p’th order) of products of Toeplitz matrices. Such products arise frequently in the analysis of stationary time series and in the development of asymptotic expansions. The elements of the matrices are Fourier transforms of functions which we allow to be bounded, unbounded, or even to vanish on [−π,π], thereby including important cases such as the spectral functions of fractional processes. Error rates are also given in the case in which the matrix product involves inverse matrices. The rates are sharp up to an arbitrarily small ε > 0. The results improve on the o (1) rates obtained in earlier work for analogous products. For the p = 1 case, an explicit second order asymptotic expansion is found for a quadratic functional of the autocovariance sequences of stationary long memory time series. The order of magnitude of the second term in this expansion is shown to depend on the long memory parameters. It is demonstrated that the pole in the Þrst order approximation is removed by the second order term, which provides a substantially improved approximation to the original functional.