  This paper develops a new direct approach to approximating suprema of general
empirical processes by a sequence of suprema of Gaussian processes, without
taking the route of approximating whole empirical processes in the sup-norm. We
prove an abstract approximation theorem applicable to a wide variety of
statistical problems, such as construction of uniform confidence bands for
functions. Notably, the bound in the main approximation theorem is
nonasymptotic and the theorem allows for functions that index the empirical
process to be unbounded and have entropy divergent with the sample size. The
proof of the approximation theorem builds on a new coupling inequality for
maxima of sums of random vectors, the proof of which depends on an effective
use of Stein's method for normal approximation, and some new empirical process
techniques. We study applications of this approximation theorem to local and
series empirical processes arising in nonparametric estimation via kernel and
series methods, where the classes of functions change with the sample size and
are non-Donsker. Importantly, our new technique is able to prove the Gaussian
approximation for the supremum type statistics under weak regularity
conditions, especially concerning the bandwidth and the number of series
functions, in those examples.
