  We study the mismatched successive refinement problem where one uses Gaussian
codebooks to compress an arbitrary memoryless source with successive minimum
Euclidean distance encoding under the quadratic distortion measure.
Specifically, we derive achievable refined asymptotics under both the joint
excess-distortion probability (JEP) and the separate excess-distortion
probabilities (SEP) criteria. For both second-order and moderate deviations
asymptotics, we consider two types of codebooks: the spherical codebook where
each codeword is drawn independently and uniformly from the surface of a sphere
and the i.i.d. Gaussian codebook where each component of each codeword is drawn
independently from a Gaussian distribution. We establish the achievable
second-order rate-region under JEP and we show that under SEP any memoryless
source satisfying mild moment conditions is strongly successively refinable.
When specialized to a Gaussian memoryless source (GMS), our results provide an
alternative achievability proof with specific code design. We show that under
JEP and SEP, the same moderate deviations constant is achievable. For large
deviations asymptotics, we only consider the i.i.d. Gaussian codebook since the
i.i.d. Gaussian codebook has better performance than the spherical codebook in
this regime for the one layer mismatched rate-distortion problem (Zhou, Tan,
Motani, TIT, 2019). We derive achievable exponents of both JEP and SEP and
specialize our results to a GMS, which appears to be a novel result of
independent interest.
