  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 Lipschitz functions, encompassing and
improving on all previous results in the literature. When specialized to the
setting considered by Rio (1994), and certain constraints on the function class
hold, our result improves the approximation rate $n^{-1/(2d)}$ to
$n^{-1/\max\{d,2\}}$, up to the same $\operatorname{polylog} n$ term, where $n$
denotes the sample size. Remarkably, we establish a valid uniform Gaussian
strong approximation at the optimal 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 a class of
multiplicative separable empirical processes indexed by Lipschitz functions,
which address some outstanding problems in the literature (Chernozhukov et al.,
2014, Section 3). In addition, two other uniform Gaussian strong approximation
results are presented for settings where the function class takes the form of a
sequence of Haar basis based on generalized quasi-uniform partitions. We
demonstrate the improvements and usefulness of our new strong approximation
results with several statistical applications to nonparametric density and
regression estimation.
