Successive refinement of information: characterization of the achievable rates

Let R(·) be the rate-distortion function. Assume that we want to describe a source with distortion no larger than Δ₁ . From the rate-distortion theory we know that we need to do so at a rate R₁ no smaller than R(Δ₁) [bits/symbol]. If it turns out that a more accurate description at distortion Δ₂, Δ₂<Δ₁ , is desirable, one can transmit additional information at some rate ΔR. What is the minimal value for ΔR? More generally, which are the achievable (R₁, R₂) pairs for which, with high probability, one can describe the source at rate R₁ and incur distortion not exceeding Δ₁ and refine this description at a rate R₂-R₁ obtaining a final distortion not exceeding Δ₂? The achievable region containing those pairs is characterized. An interpretation of Equitz and Cover´s Markov condition is given. The Markov condition characterizes those cases for which (R(Δ₁), R(Δ₂)) is an achievable pair