Rate-distortion functions for source coding when side information with unknown delay may be present

In this paper, we consider a lossy source coding problem with an encoder and two decoders, in which side information is available at one of the decoders with an unknown delay. We assume that the maximum of delay is known to among the encoder and two decoders. In this coding problem, we show upper and lower bounds on the rate-distortion (RD) function, where the RD function is the infimum of rates of codes of which the distortion between the source sequence and the reproduction sequence satisfies a certain distortion level. We also show that the upper bound coincides with the lower bound when the maximum of delay per block length converges to a constant. Furthermore, we show a condition such that the RD function is strictly larger than that for the case of no delay.