  In this paper, we revisit two multi-terminal lossy source coding problems:
the lossy source coding problem with side information available at the encoder
and one of the two decoders, which we term as the Kaspi problem (Kaspi, 1994),
and the multiple description coding problem with one semi-deterministic
distortion measure, which we refer to as the Fu-Yeung problem (Fu and Yeung,
2002). For the Kaspi problem, we first present the properties of optimal test
channels. Subsequently, we generalize the notion of the distortion-tilted
information density for the lossy source coding problem to the Kaspi problem
and prove a non-asymptotic converse bound using the properties of optimal test
channels and the well-defined distortion-tilted information density. Finally,
for discrete memoryless sources, we derive refined asymptotics which includes
the second-order, large and moderate deviations asymptotics. In the converse
proof of second-order asymptotics, we apply the Berry-Esseen theorem to the
derived non-asymptotic converse bound. The achievability proof follows by first
proving a type-covering lemma tailored to the Kaspi problem, then properly
Taylor expanding the well-defined distortion-tilted information densities and
finally applying the Berry-Esseen theorem. We then generalize the methods used
in the Kaspi problem to the Fu-Yeung problem. As a result, we obtain the
properties of optimal test channels for the minimum sum-rate function, a
non-asymptotic converse bound and refined asymptotics for discrete memoryless
sources. Since the successive refinement problem is a special case of the
Fu-Yeung problem, as a by-product, we obtain a non-asymptotic converse bound
for the successive refinement problem, which is a strict generalization of the
non-asymptotic converse bound for successively refinable sources (Zhou, Tan and
Motani, 2017).
