The output distribution of good lossy source codes

This paper provides a necessary condition good rate-distortion codes must satisfy. Specifically, it is shown that as the blocklength increases, the distribution of the input given the output of a good lossy code converges to the distribution of the input given the output of the joint distribution achieving the rate-distortion function, in terms of the normalized conditional relative entropy. The result holds for stationary ergodic sources with subadditive distortion measures, both for fixed-length and variable-length compression. A similar necessary condition is given for lossy joint source-channel coding.