  This paper is concerned with finite sample approximations to the supremum of
a non-degenerate $U$-process of a general order indexed by a function class. We
are primarily interested in situations where the function class as well as the
underlying distribution change with the sample size, and the $U$-process itself
is not weakly convergent as a process. Such situations arise in a variety of
modern statistical problems. We first consider Gaussian approximations, namely,
approximate the $U$-process supremum by the supremum of a Gaussian process, and
derive coupling and Kolmogorov distance bounds. Such Gaussian approximations
are, however, not often directly applicable in statistical problems since the
covariance function of the approximating Gaussian process is unknown. This
motivates us to study bootstrap-type approximations to the $U$-process
supremum. We propose a novel jackknife multiplier bootstrap (JMB) tailored to
the $U$-process, and derive coupling and Kolmogorov distance bounds for the
proposed JMB method. All these results are non-asymptotic, and established
under fairly general conditions on function classes and underlying
distributions. Key technical tools in the proofs are new local maximal
inequalities for $U$-processes, which may be useful in other problems. We also
discuss applications of the general approximation results to testing for
qualitative features of nonparametric functions based on generalized local
$U$-processes.
