Pointwise lossy source coding theorem for sources with memory

We investigate the minimum pointwise redundancy of variable length lossy source codes operating at fixed distortion for sources with memory. The redundancy is defined by l_n(X₁ⁿ) - nR(D), where l_n(X₁ⁿ) is the code length at block size n and R(D) is the rate distortion function. We restrict ourselves to the case where R(D) can be calculated, namely the cases where the Shannon lower bound to R(D) holds with equality. In this case, for balanced distortion measures, we provide a pointwise lower bound to the code length sequence in terms of the entropy density process. We show that the minimum coding variance with distortion is lower bounded by the minimum lossless coding variance, and is non-zero unless the entropy density is deterministic. We also examine lossy coding in the presence of long range dependence, showing the existence of information sources for which long range dependence persists under any codec operating at the Shannon lower bound with fixed distortion.