The Distortion-Rate Function for Nonergodic Sources

The distortion rate function D (R) is defined as an infimum of distortion with respect to a mutual information constraint. The usual coding theorems assert that, for ergodic souces, D (R) is equal to (R), the least distortion attainable by block codes of rate R. If a source has ergodic components \\ with weighting measure dw (), it has been shown by Gray and Davisson that (R) is the integral of the components _ (R) with respect to dw (). We show that D (R) is the infimum of the integrals of D_ (R_) where the integral of R_ is R. Our method of proof also gives a formula for the d-distance in terms of ergodic components and shows that D (R) = D' (R), which is defined as the infimum of distortion subject to an entropy constraint.