This paper presents new uniform Gaussian strong approximations for empirical processes indexed by classes of functions based on $d$-variate random vectors ($d\geq1$). First, a uniform Gaussian strong approximation is established for general empirical processes indexed by possibly Lipschitz functions, improving on previous results in the literature. In the setting considered by Rio (1994), and if the function class is Lipschitzian, our result improves the approximation rate $n^{-1/(2d)}$ to $n^{-1/\max\{d,2\}}$, up to a $\operatorname{polylog}(n)$ term, where $n$ denotes the sample size. Remarkably, we establish a valid uniform Gaussian strong approximation at the rate $n^{-1/2}\log n$ for $d=2$, which was previously known to be valid only for univariate ($d=1$) empirical processes via the celebrated Hungarian construction (Koml\'os et al., 1975). Second, a uniform Gaussian strong approximation is established for multiplicative separable empirical processes indexed by possibly Lipschitz functions, which addresses some outstanding problems in the literature (Chernozhukov et al., 2014, Section 3). Finally, two other uniform Gaussian strong approximation results are presented when the function class is a sequence of Haar basis based on quasi-uniform partitions. Applications to nonparametric density and regression estimation are discussed.

翻译：本文针对基于$d$维随机向量（$d\geq1$）的函数类索引的经验过程，提出了新的均匀高斯强逼近方法。首先，为可能具有Lipschitz性质的函数索引的一般经验过程建立了均匀高斯强逼近，改进了文献中的已有结果。在Rio (1994) 所考虑的设定下，若函数类具有Lipschitz性质，我们的结果将逼近速率从$n^{-1/(2d)}$提升至$n^{-1/\max\{d,2\}}$（相差一个$\operatorname{polylog}(n)$因子），其中$n$表示样本量。值得注意的是，对于$d=2$的情形，我们建立了以$n^{-1/2}\log n$速率成立的均匀高斯强逼近，而此前仅通过著名的匈牙利构造（Komlós et al., 1975）已知该速率对单变量（$d=1$）经验过程有效。其次，针对可能具有Lipschitz性质的函数索引的可乘可分经验过程建立了均匀高斯强逼近，这解决了文献中的一些遗留问题（Chernozhukov et al., 2014, Section 3）。最后，当函数类是基于拟均匀分割的Haar基序列时，给出了另外两个均匀高斯强逼近结果。文中还讨论了在非参数密度估计与回归估计中的应用。