We present nonasymptotic concentration inequalities for sums of independent and identically distributed random variables that yield asymptotic strong Gaussian approximations of Koml\'os, Major, and Tusn\'ady (KMT) [1975,1976]. The constants appearing in our inequalities are either universal or explicit, and thus as corollaries, they imply distribution-uniform generalizations of the aforementioned KMT approximations. In particular, it is shown that uniform integrability of a random variable's $q^{\text{th}}$ moment is both necessary and sufficient for the KMT approximations to hold uniformly at the rate of $o(n^{1/q})$ for $q > 2$ and that having a uniformly lower bounded Sakhanenko parameter -- equivalently, a uniformly upper-bounded Bernstein parameter -- is both necessary and sufficient for the KMT approximations to hold uniformly at the rate of $O(\log n)$. Instantiating these uniform results for a single probability space yields the analogous results of KMT exactly.

翻译：本文针对独立同分布随机变量和提出了非渐近集中不等式，这些不等式可导出Komlós、Major与Tusnády（KMT）[1975,1976]的渐近强高斯逼近。所呈现不等式中的常数具有普适性或显式表达，因此作为推论，它们意味着对前述KMT逼近的分布均匀推广。特别地，研究证明：随机变量$q^{\text{th}}$阶矩的一致可积性（其中$q > 2$）是KMT逼近以$o(n^{1/q})$速率均匀成立的充要条件；而Sakhanenko参数的一致下有界性——等价于Bernstein参数的一致上有界性——是KMT逼近以$O(\log n)$速率均匀成立的充要条件。将这些均匀结果实例化到单个概率空间，可精确得到KMT的对应结论。