  This paper studies the third-order characteristic of nonsingular discrete
memoryless channels and the Gaussian channel with a maximal power constraint.
The third-order term in our expansions employs a new quantity here called the
\emph{channel skewness}, which affects the approximation accuracy more
significantly as the error probability decreases. For the Gaussian channel,
evaluating Shannon's (1959) random coding and sphere-packing bounds in the
central limit theorem (CLT) regime enables exact computation of the channel
skewness. For discrete memoryless channels, this work generalizes Moulin's
(2017) bounds on the asymptotic expansion of the maximum achievable message set
size for nonsingular channels from the CLT regime to include the moderate
deviations (MD) regime, thereby refining Altu\u{g} and Wagner's (2014) MD
result. For an example binary symmetric channel and most practically important
$(n, \epsilon)$ pairs, including $n \in [100, 500]$ and $\epsilon \in
[10^{-10}, 10^{-1}]$, an approximation up to the channel skewness is the most
accurate among several expansions in the literature. A derivation of the
third-order term in the type-II error exponent of binary hypothesis testing in
the MD regime is also included; the resulting third-order term is similar to
the channel skewness.
