Achievable Second-Order Asymptotics for Successive Refinement Using Gaussian Codebooks

We study the mismatched successive refinement problem where one uses a fixed code to compress an arbitrary source with random Gaussian codebooks and minimum Euclidean distance encoding in a successive manner. Specifically, we generalize the mismatched rate-distortion framework by Lapidoth (T-IT, 1997) to the successive refinement setting and derive the achievable second-order asymptotics. Our result implies that any source that satisfies a mild moment constraint is successive refinable under our code. Furthermore, our proof, when specialized to a Gaussian memoryless source, provides an alternative achievability proof with structured codebooks for the successive refinement problem, which was studied by Zhou, Tan, Motani (T-IT, 2018) where a covering lemma without specifying the locations of codewords was used.