Error exponents for successive refinement by partitioning

Given a discrete memoryless source (DMS) with probability mass function P, we seek first an asymptotically optimal description of the source with distortion not exceeding Δ₁, followed by an asymptotically optimal refined description with distortion not exceeding Δ₂<Δ₁. The rate-distortion function for successive refinement by partitioning, denoted R(P, Δ₁ , Δ₂), is the overall optimal rate of these descriptions obtained via a two-step coding process. We determine the error exponents for this two-step coding process, namely, the negative normalized asymptotic log likelihoods of the event that the distortion in either step exceeds its prespecified acceptable value, and of the conditional event that the distortion in the second step exceeds the prespecified value given the rate and distortion of the code for the first step. We show that even when the rate-distortion functions for one- and two-step coding coincide, the error exponent in the former case may exceed those in the latter