Compressing a long range dependent renewal process

Analysis of variable bit-rate video data has shown that long range dependence persists across a wide variety of codecs. While codecs are generally lossy, one may conjecture, as a partial explanation for this fact, that there exist information sources for which any lossless code results in a bit-rate process that eventually dominates a long range dependent random process. We prove this to be true for discrete time long range dependent renewal processes under a mild technical assumption.