On the asymptotic tightness of the Shannon lower bound

New results are proved on the convergence of the Shannon (1959) lower bound to the rate distortion function as the distortion decreases to zero. The key convergence result is proved using a fundamental property of informational divergence. As a corollary, it is shown that the Shannon lower bound is asymptotically tight for norm-based distortions, when the source vector has a finite differential entropy and a finite α th moment for some α>0, with respect to the given norm. Moreover, we derive a theorem of Linkov (1965) on the asymptotic tightness of the Shannon lower bound for general difference distortion measures with more relaxed conditions on the source density. We also show that the Shannon lower bound relative to a stationary source and single-letter difference distortion is asymptotically tight under very weak assumptions on the source distribution