Concentration inequalities for the sample mean, like those due to Bernstein and Hoeffding, are valid for any sample size but overly conservative, yielding confidence intervals that are unnecessarily wide. The central limit theorem (CLT) provides asymptotic confidence intervals with optimal width, but these are invalid for all sample sizes. To resolve this tension, we develop new computable concentration inequalities with asymptotically optimal size, finite-sample validity, and sub-Gaussian decay. These bounds enable the construction of efficient confidence intervals with correct coverage for any sample size. We derive our inequalities by tightly bounding the Hellinger distance, Stein discrepancy, non-uniform Kolmogorov distance, and Wasserstein distance to a Gaussian, and, as a byproduct, we obtain the first explicit bounds for the Hellinger CLT.

翻译：针对样本均值的集中不等式（如Bernstein和Hoeffding不等式）虽适用于任意样本量，但过于保守，导致置信区间过宽。中心极限定理（CLT）虽能提供具有最优宽度的渐近置信区间，但对于所有样本量并非严格成立。为解决这一矛盾，我们提出了新型可计算的集中不等式，其具备渐近最优尺度、有限样本有效性及次高斯衰减特性。这些界使得对于任意样本量均可构建具有正确覆盖概率的高效置信区间。我们的推导通过对Hellinger距离、Stein散度、非均匀Kolmogorov距离和Wasserstein距离的高斯近似进行严格约束而实现，并由此首次得到Hellinger中心极限定理的显式界。