  In this paper, we revisit the discrete lossy Gray-Wyner problem. In
particular, we derive its optimal second-order coding rate region, its error
exponent (reliability function) and its moderate deviations constant under mild
conditions on the source. To obtain the second-order asymptotics, we extend
some ideas from Watanabe's work (2015). In particular, we leverage the
properties of an appropriate generalization of the conditional
distortion-tilted information density, which was first introduced by Kostina
and Verd\'u (2012). The converse part uses a perturbation argument by Gu and
Effros (2009) in their strong converse proof of the discrete Gray-Wyner
problem. The achievability part uses two novel elements: (i) a generalization
of various type covering lemmas; and (ii) the uniform continuity of the
conditional rate-distortion function in both the source (joint) distribution
and the distortion level. To obtain the error exponent, for the achievability
part, we use the same generalized type covering lemma and for the converse, we
use the strong converse together with a change-of-measure technique. Finally,
to obtain the moderate deviations constant, we apply the moderate deviations
theorem to probabilities defined in terms of information spectrum quantities.
