A judicious application of the Berry-Esseen theorem via suitable Augustin information measures is demonstrated to be sufficient for deriving the sphere packing bound with a prefactor that is $\mathit{\Omega}\left(n^{-0.5(1-E_{sp}'(R))}\right)$ for all codes on certain families of channels -- including the Gaussian channels and the non-stationary Renyi symmetric channels -- and for the constant composition codes on stationary memoryless channels. The resulting non-asymptotic bounds have definite approximation error terms. As a preliminary result that might be of interest on its own, the trade-off between type I and type II error probabilities in the hypothesis testing problem with (possibly non-stationary) independent samples is determined up to some multiplicative constants, assuming that the probabilities of both types of error are decaying exponentially with the number of samples, using the Berry-Esseen theorem.

中文翻译：

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特定对称假设下细化球填充界的简单推导

Berry-Esseen 定理通过合适的奥古斯丁信息度量的明智应用被证明足以推导出具有 $\mathit{\Omega}\left(n^{-0.5(1-E_{ sp}'(R))}\right)$ 用于某些通道系列上的所有代码——包括高斯通道和非平稳 Renyi 对称通道——以及用于平稳无记忆通道上的恒定组合码。由此产生的非渐近边界具有确定的近似误差项。作为可能本身感兴趣的初步结果，在具有（可能是非平稳）独立样本的假设检验问题中，I 类和 II 类错误概率之间的权衡取决于一些乘法常数，