  In 2008 Arikan proposed polar coding [arXiv:0807.3917] which we summarize as
follows: (a) From the root channel $W$ synthesize recursively a series of
channels $W_N^{(1)},\dotsc,W_N^{(N)}$. (b) Select sophisticatedly a subset $A$
of synthetic channels. (c) Transmit information using synthetic channels
indexed by $A$ and freeze the remaining synthetic channels.
  Arikan gives each synthetic channel a score (called the Bhattacharyya
parameter) that determines whether it should be selected or frozen. As $N$
grows, a majority of the scores are either very high or very low, i.e., they
polarize. By characterizing how fast they polarize, Arikan showed that polar
coding is able to produce a series of codes that achieve capacity on symmetric
binary-input memoryless channels.
  In measuring how the scores polarize the relation among block length, gap to
capacity, and block error probability are studied. In particular, the error
exponent regime fixes the gap to capacity and varies the other two. The scaling
exponent regime fixes the block error probability and varies the other two. The
moderate deviation regime varies all three factors at once.
  The latest result [arxiv:1501.02444, Theorem 7] in the moderate deviation
regime does not imply the scaling exponent regime as a special case. We give a
result that does. (See Corollary 8.)
