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 δ(R), the least distortion attainable by block codes of rate R. If a source has ergodic components \θ\ with weighting measure dw(θ), it has been shown by Gray and Davisson that δ(R) is the integral of the components δ_θ(R) with respect to dw(θ). We show that $D(R)$ is the infimum of the integrals of D_θ(R_θ) where the integral of R_θ is R. Our method of proof also gives a formula for the d̄-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.