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Recently, Altug and Wagner posed a question regarding the optimal behavior of the probability of error when channel coding rate converges to the capacity sufficiently slowly. They gave a sufficient condition for the Discrete Memory-less Channel (DMC) to satisfy a Moderate Deviation Property (MDP) with the constant equal to the channel dispersion. Their sufficient condition excludes some practically interesting channels, such as the binary erasure channel and the Z-channel. The authors extend their result in two directions. First, they show that a DMC satisfies MDP if and only if its channel dispersion is nonzero. Second, they prove that the AWGN channel also satisfies MDP with a constant equal to the channel dispersion.
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