Inverse moment bounds for sample autocovariance matrices based on detrended time series and their applications

In this paper, we assume that observations are generated by a linear regression model with short- or long-memory dependent errors. We establish inverse moment bounds for knkn-dimensional sample autocovariance matrices based on the least squares residuals (also known as the detrended time series), where kn≪nkn≪n, kn→∞kn→∞ and n is the sample size. These results are then used to derive the mean-square error bounds for the finite predictor coefficients of the underlying error process. Based on the detrended time series, we further estimate the inverse of the n  -dimensional autocovariance matrix, Rn−1, of the error process using the banded Cholesky factorization. By making use of the aforementioned inverse moment bounds, we obtain the convergence of moments of the difference between the proposed estimator and Rn−1 under spectral norm.