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The generalized Shannon lower bound to the rate-distortion function for stationary sources with memory is extended to a wide class of distortion measures involving no symmetry conditions. The lower bound is a reasonably simple function of the entropy and marginal probabilities of the source and the per-letter distortion measure. Sufficient conditions only slightly less general than necessary conditions are given for the existence of a strictly positive cutoff distortion such that for . The sufficient conditions are the most general to date and include all previously known examples. This provides a nearly complete resolution of the question of when the Shannon-type lower bound to the rate-distortion function of a source with memory is tight. The results are applied to generalize earlier results for balanced distortion measures and Markov sources to nonbalanced distortion measures and wide-sense Markov sources. As a special case, it is shown that for all finite-alphabet autoregressive sources. As an example, is evaluated for the first-order ternary autoregressive source for a balanced (Hamming) and a nonbalanced (modular distance) distortion measure. A simple lower bound to is derived for this example.