Properties of generalized Levinson-Durbin-Whittle sequences

For a discrete time, second-order stationary process the Levinson-Durbin recursion is used to determine best fitting one-step-ahead linear autoregressive predictors of successively increasing order, best in the sense of minimizing the mean square error. Whittle [1963. On the fitting of multivariate autoregressions, and the approximate canonical factorization of a spectral density matrix. Biometrika 50, 129-134] generalized the recursion to the case of vector autoregressive processes. The recursion defines what is termed a Levinson-Durbin-Whittle sequence, and a generalized Levinson-Durbin-Whittle sequence is also defined. Generalized Levinson-Durbin-Whittle sequences are shown to satisfy summation formulas which generalize summation formulas satisfied by binomial coefficients. The formulas can be expressed in terms of the partial correlation sequence, and they assume simple forms for time-reversible processes. The results extend comparable formulas obtained in Shaman [2007. Generalized Levinson-Durbin sequences, binomial coefficients and autoregressive estimation. Working paper] for univariate processes.