  The rate-distortion saddle-point problem considered by Lapidoth (1997)
consists in finding the minimum rate to compress an arbitrary ergodic source
when one is constrained to use a random Gaussian codebook and minimum
(Euclidean) distance encoding is employed. We extend Lapidoth's analysis in
several directions in this paper. Firstly, we consider refined asymptotics. In
particular, when the source is stationary and memoryless, we establish the
second-order, moderate, and large deviation asymptotics of the problem.
Secondly, by "random Gaussian codebook", Lapidoth referred to a collection of
random codewords, each of which is drawn independently and uniformly from the
surface of an $n$-dimensional sphere. To be more precise, we term this as a
spherical codebook. We also consider i.i.d.\ Gaussian codebooks in which each
random codeword is drawn independently from a product Gaussian distribution. We
derive the second-order, moderate, and large deviation asymptotics when i.i.d.\
Gaussian codebooks are employed. In contrast to the recent work on the channel
coding counterpart by Scarlett, Tan and Durisi (2017), the dispersions for
spherical and i.i.d.\ Gaussian codebooks are identical. The ensemble
excess-distortion exponents for both spherical and i.i.d.\ Gaussian codebooks
are established for all rates. Furthermore, we show that the i.i.d.\ Gaussian
codebook has a strictly larger excess-distortion exponent than its spherical
counterpart for any rate greater than the ensemble rate-distortion function
derived by Lapidoth.
