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：样本均值的集中不等式，如伯恩斯坦和霍夫丁不等式，虽然适用于任意样本量，但过于保守，导致置信区间不必要地宽。中心极限定理（CLT）提供了具有最优宽度的渐近置信区间，但这些区间并非对所有样本量都有效。为解决这一矛盾，我们开发了新的可计算集中不等式，具有渐近最优大小、有限样本有效性以及次高斯衰减。这些界限使得能够构建适用于任意样本量的正确覆盖的高效置信区间。我们通过紧密界定与高斯的Hellinger距离、Stein差异、非均匀Kolmogorov距离和Wasserstein距离来推导这些不等式，并作为副产品，首次获得了Hellinger CLT的显式界限。