  Consider a source that produces independent copies of a triplet of jointly
distributed random variables, $\{X_{i},Y_{i},Z_{i}\}_{i=1}^{\infty}$. The
process $\{X_{i}\}$ is observed at the encoder, and is supposed to be
reproduced at two decoders, where $\{Y_{i}\}$ and $\{Z_{i}\}$ are observed, in
either a causal or non-causal manner. The communication between the encoder and
the decoders is carried in two successive stages. In the first stage, the
transmission is available to both decoders and the source is reconstructed
according to the received bit-stream and the individual side information (SI).
In the second stage, additional information is sent to both decoders and the
source reconstructions are refined according to the transmissions at both
stages and the available SI. It is desired to find the necessary and sufficient
conditions on the communication rates between the encoder and decoders, so that
the distortions incurred (at each stage) will not exceed given thresholds. For
the case of non-degraded causal SI at the decoders, an exact single-letter
characterization of the achievable region is derived for the case of pure
source-coding. Then, for the case of communication carried over independent
DMS's with random states known causally/non-causally at the encoder and with
causal SI about the source at the decoders, a single-letter characterization of
all achievable distortion in both stages is provided and it is shown that the
separation theorem holds. Finally, for non-causal degraded SI, inner and outer
bounds to the achievable rate-distortion region are derived. These bounds are
shown to be tight for certain cases of reconstruction requirements at the
decoders, thereby shading some light on the problem of successive refinement
with non-degraded SI at the decoders.
