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.

翻译：本文研究了无奇异离散无记忆信道和具有最大功率约束的高斯信道在三阶特性上的表现。在我们的展开式中，三阶项采用了一个新量——称为**信道偏度**，该量随着错误概率的降低对近似精度的影响越来越显著。对于高斯信道，通过评估香农（1959）在中心极限定理（CLT）机制下的随机编码和球堆积界，可以精确计算信道偏度。对于离散无记忆信道，本文将穆兰（2017）关于无奇异信道最大可达消息集大小渐近展开的界从CLT机制推广到包括中等偏差（MD）机制，从而完善了阿尔图与瓦格纳（2014）的MD结果。以二进制对称信道为例，对于大多数实际重要的$(n, \epsilon)$组合（包括$n \in [100, 500]$和$\epsilon \in [10^{-10}, 10^{-1}]$），在文献中的多种展开式中，精确到信道偏度的近似最为准确。此外，本文还推导了MD机制下二元假设检验中第二类错误指数的三阶项，所得的三阶项与信道偏度相似。