VARIABLE-RATE SOURCE-CODING THEOREMS FOR STATIONARY NONERGODIC SOURCES

For a stationary ergodic source, the source coding theorem and its con verse imply that the optimal performance theoretically achievable by a fixed-rate or variable-rate block quantizer is equal to the distortio n-rate function, which is defined as the infimum of an expected distor tion subject to a mutual information constraint. For a stationary none rgodic source, however, the distortion-rate function cannot in general be achieved arbitrarily closely by a fixed-rate block code. We show, though, that for any stationary nonergodic source with a Polish alphab et, the distortion-rate function can be achieved arbitrarily closely b y a variable-rate block code. We also show that the distortion-rate fu nction of a stationary nonergodic source has a decomposition as the av erage of the distortion-rate functions of the source's stationary ergo dic components, where the average is taken over points on the componen t distortion-rate functions having the same slope. These results exten d previously known results for finite alphabets.