ON THE ASYMPTOTIC TIGHTNESS OF THE SHANNON LOWER-BOUND

New results are proved on the convergence of the Shannon lower bound t o the rate distortion function as the distortion decreases to zero. Th e key convergence result is proved using a fundamental property of inf ormational divergence. As a corollary, it is shown that the Shannon lo wer bound is asymptotically tight for norm-based distortions, when the source vector has a finite differential entropy and a finite alpha th moment for some alpha > 0, with respect to the given norm. Moreover, we derive a theorem of Linkov on the asymptotic tightness of the Shann on lower bound for general difference distortion measures with more re laxed conditions on the source density. We also show that the Shannon lower bound relative to a stationary source and single-letter differen ce distortion is asymptotically tight under very weak assumptions on t he source distribution.