Rate-Distortion Functions for Nonstationary Gaussian Autoregressive Processes

The Shannon rate-distortion function R(D) of a random process provides a lower bound to the minimal average distortion given a constraint on the average rate. When a positive source coding theorem with a fidelity criterion applies, the lower bound is achievable in the limit of large block length and hence R(D) characterizes the optimal performance for source coding or lossy data compression. The source coding theorem for possibly nonstationary Gaussian autoregressive sources was established over three decades ago, but two apparently different formulas for R(D) have appeared in the literature, resulting in long standing confusion about which is correct. There has also been related confusion about the asymptotic eigenvalue distributions of the inverse covariance matrices of such processes. We here establish the equality of the two formulas under fairly general conditions and clarify the confusion regarding asymptotic eigenvalue distributions.