Kaspi problem revisited: Non-Asymptotic converse bound and second-order asymptotics

In this paper, we revisit 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). For the Kaspi problem, we first present the properties of optimal test channels for the rate-distortion function. Subsequently, we generalize the notion of 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, we derive the exact second-order coding rate of the Kaspi problem for discrete memoryless sources.